Jiayu Li

Xi Zhang

Rugang Ye

Xiaowu Chen

Bin Xu

Congwen Liu

Bailin Song

Guangbin Ren

Zuoqing Wang

Xiaowei Xu

Tiren Huang

etc... Ils Qin Li

Damien Bouloc

Hao Ding

Anton Eloy

Sergio Lugo Gutierrez

TatDat To, Tao Xiong etc... Songzi Li

Yi Xu

Guohuan Fang

Jingy- Ong Qiu

Jun Zhu

Dekai Wang

Liangjun Zhang

Wei Weng

Bin Wei

Xiaohan Deng

Zhenghuan Jia

etc... Gao

de la technologie de Chine

Keywords: Transport optimal, équation de Monge-Ampère, fonction c-convexe, tenseur de MTW, condition de MTW, convexité du domaine d Optimal transport map, Monge-Ampère equation, c-convex function, MTW tensor, MTW condition, convexity of injectivity domain, maximum principle, continuity method. iii

Dans cette thèse on s'intéressons à la régularité de l'application du transport optimal sur des variétés riemanniennes compactes.

Dans le premier chapitre, on rappelle certaines définitions sur une variété riemannienne.

Dans le deuxième chapitre, on décrit la variation de la courbure sur des géodésiques.

Dans le troisième chapitre, on étudie le tenseur de MTW sur une variété riemannienne compacte. On montre qu'une condition de MTW améliorée est satisfaite sur une variété presque sphérique. La preuve consiste à une analyse minutieuse, combinée avec les arguments de perturbation sur des sphères.

Dans le quatrième chapitre, on étudie le comportement de l'inverse de la matrice Hessienne de la distance au carré.

Dans le cinquième chapitre, on prouve la régularité du transport optimale sur deux classes des variétés riemanniennes compactes-des variétés presque sphériques et des produits riemanniens des variétés presque sphériques.

Dans le dernier chapitre, on déscrit quelques perspectives sur le transport optimal dans la littérature.

Contents

Remerciements v

Synthèse de la thèse ix 0.1 Le problème du transport optimal . . . . . . . . . . . . . . . . . . . . . ix 0.2 L'existence de l'application du transport optimal . . . . . . . . . . . . . x 0.3 La régularité de l'application du transport optimal . . . . . . . . . . . . xi 0.3.1 L'équation du transport optimal . . . . . . . . . . . . . . . . . . xi

Synthèse de la thèse

Dans cette thèse, on se concentre sur la régularité de l'application du transport optimal. Ce sujet a été amplement étudié ces dernières années. On prouve la régularité de l'application du transport optimale sur deux classes de variétés riemanniennes compactesdes variétés presque sphériques et des produits riemanniens des variétés presque sphériques.

Le problème du transport optimal

Le transport optimal est un sujet ancien. Il est étudié pour la première fois par Monge en 1781 [START_REF] Monge | Mémoire sur la théorie des déblais et remblais[END_REF] avec le coût de la distance euclidienne. Depuis, il est apparu dans de nombreux domaines tels que la théorie de probabilité, l'économie, l'optimisation, la météorologie, etc... L'introduction générale à la théorie du transport optimal peut être trouvée dans des livres [START_REF] Villani | Topics in optimal transportation[END_REF] [START_REF] Villani | Optimal transport, old and new[END_REF].

Le problème du transport optimal s'exprime comme suit: soient (X, µ 0 ) et (Y, µ 1 ) deux espaces métriques avec des mesures de probabilité µ 0 et µ 1 respectivement. Soit c : X × Y → R une fonction du coût. Le problème du transport optimal consiste à minimiser la fonctionnelle du coût total Cost(G) = ∫ X c(x, G(x))dµ 0 parmi toutes les applications mesurables G : X → Y , telles que G # µ 0 = µ 1 , cela signifie que pour tout ensemble E ⊂ X mesurable, on a µ 1 (E) = µ 0 (G -1 (E)).

Les minimiseurs sont appelés les applications du transport optimal.

L'existence de l'application du transport optimal n'est pas triviale. D'une part, il peut y avoir aucune application tel que G # µ 0 = µ 1 . Par exemple, quand µ 0 est égale à la mesure de Dirac alors que µ 1 ne l'est pas. D'autre part, le problème est non linéaire.

Cent soixante ans plus tard après Monge, Kantorovich [START_REF] Kantorovich | On the transfer of masses[END_REF] a réduit le problème cidessus à un programme linéaire à dimension infinie. Plus précisément, on cherche une mesure de probabilité µ sur X × Y tel que ∫

L'existence de l'application du transport optimal

Avant d'introduire l'existence de l'application du transport optimal, on donne d'abord la définition de la c-convexité [START_REF] Villani | Topics in optimal transportation[END_REF]. Définition 0.1. (Fonction c-convexe) Soit c : X × Y → R une fonction. La fonction u : X → R ∪ {∞} est c-convexe, si pour tout x 0 ∈ X, il existe un y 0 ∈ Y tel que ∀x ∈ X, u(x) ≥ u(x 0 ) + c(x 0 , y 0 )c(x, y 0 ). La c-sousdifférentiel de la fonction u au point x 0 est définie comme ∂ c u(x 0 ) = {y 0 ∈ Y : ∀x ∈ X, u(x) ≥ u(x 0 ) + c(x 0 , y 0 )c(x, y 0 )}. Maintenant, on décrit l'existence de l'application du transport optimal. Lorsque c(x, y) = |x -y|, Sudakov [START_REF] Sudakov | Geometric problems in the theory of infinite dimensional probability distributions[END_REF], Evans-Gangbo [START_REF] Evans | Differential equations methods for the MongeĺCKantorovich mass transfer problem[END_REF], Caffarelli-Feldman-Mccann [START_REF] Caffarelli | Constructing optimal maps for Mongès transport problem as a limit of strictly convex costs[END_REF], Trudinger-Wang [START_REF] Trudinger | On the Monge mass transfer problem[END_REF] ont démontré l'existence de l'application du transport optimal. En général, l'application du transport optimal n'est pas unique. Lorsque c(x, y) = 1 2 |x-y| 2 , si µ 0 est absolument continu par rapport à la mesure de Lebesgue, Brenier [START_REF] Brenier | Polar factorization and monotone rearrangement of vector-valued functions[END_REF] a montré l'existence et l'unicité de l'application du transport optimal. Il a également prouvé que l'application du transport optimal est le gradient d'une fonction convexe. Pour une fonction du coût général, l'existence de l'application du transport optimal peut être trouvée dans [START_REF] Caffarelli | Allocation maps with general cost functions[END_REF] [51] [START_REF] Villani | Topics in optimal transportation[END_REF].

Lorsque c est égal à la moitié de la distance géodésique au carré sur une variété riemannienne compacte, McCann [START_REF] Mccann | Polar factorization of maps on Riemannian manifolds[END_REF] a généralisé la théorie de Brenier. Si µ 0 est absolument continu par rapport à l'élément de volume, McCann a prouvé qu'il existe une unique application du transport optimal G. Il a également montré que G(m) = exp m (∇ m u) pour une certaine c-convexe fonction u. La fonction u est appelée le potentiel du transport optimal. Sur une variété riemannienne non-compacte, l'existence de l'application du transport optimale peut être trouvée dans [START_REF] Fathi | Optimal transportation on non-compact manifolds[END_REF]. On étudie ici la régularité de l'application du transport optimal sur une variété riemannienne compacte. On a déjà vu que l'application du transport optimal s'écrit comme G(m) = exp m (∇ m u). L'étude de la régularité de l'application du transport optimal revient à étudier la régularité du potentiel. Le potentiel u vérifie une équation elliptique complètement non-linéaire.

L'équation du transport optimal

Soient ρ 0 dvol et ρ 1 dvol deux mesures de probabilité sur une variété riemannienne compacte (M, g) à densité continue strictement positive. Si le potentiel u est de classe C 2 , alors l'application du transport optimal G est un difféomorphisme de classe C 1 .

La condition G # µ 0 = µ 1 , signifie que pour tout ensemble mesurable E ⊂ M , on a ∫

G -1 (E) ρ 0 (x)dvol(x) = ∫ E ρ 1 (y)dvol(y).
Par un changement de variable y = G(x), on en déduit ∫

G -1 (E) ρ 0 (x)dvol(x) = ∫ G -1 (E) ρ 1 (G(x))| det d x G|dvol(x). ce qui donne ∀x ∈ M, | det d x G| = ρ 0 (x) ρ 1 (G(x))
.

En utilisant la propriété de la fonction c-convexe u, on a ∀x ∈ M, ∇ x u + ∇ x c(x, G(x)) = 0.

En différenciant par rapport à x, on obtient ∀x ∈ M, det(∇ 2 x u + ∇ 2 x c(x, G(x))) = det(-∇ x,y c(x, G(x))d x G).

Encore par la propriété de la fonction c-convexe u,

∇ 2 x u + ∇ 2 x c(x, G(x)) est positif. En conséquence, det(∇ 2 x u + ∇ 2 x c(x, G(x))) = | det ∇ x,y c(x, G(x))|| det d x G| = ρ 0 (x) ρ 1 (G(x)) | det ∇ x,y c(x, G(x))|.
En vertu de exp x (-∇ x c(x, y)) = y, on en déduit

det(∇ 2 x u + ∇ 2 x c(x, G(x))) = ρ 0 (x) | det d ∇xu exp x |ρ 1 (G(x))
.

En rappelant det d ∇xu exp x > 0, on obtient l'équation

∀x ∈ M, det(∇ 2 u + ∇ 2 x c(x, G(x))) = ρ 0 (x) ρ 1 (G(x)) det d ∇u exp . ( 3
)
On donne ici quelques exemples. Dans le cas euclidien, l'équation (3) est de type Monge-Ampère généralisé det(∇ 2 u + I) = ρ 0 (x) ρ 1 (G(x)) ,

xii

CONTENTS

Lorsque M est un tore T n , l'équation (3) s'écrit det(∇ 2 u + g) = ρ 0 (x) ρ 1 (G(x))

.

Lorsque M est une sphère S n , l'équation (3) devient sin |∇u| |∇u|

(n-1) det(∇ 2 u + S(x, ∇u)) = ρ 0 (x) ρ 1 (G(x)) , où S(m, ν)(ξ) = ξ -(1 -|ν| cot |ν|)(ξ -g m (ξ, ν |ν| ) ν |ν| ).

Le tenseur de Ma-Trudinger-Wang

L'application du transport optimal n'est pas forcément continue ou lisse. Afin de garantir une certaine régularité, des hypothèses supplémentaires sont nécessaires. Dans l'espace euclidien, Ma-Trudinger-Wang [START_REF] Ma | Regularity of potential functions of the optimal transportation problem[END_REF] ont introduit une quantité en utilisant les dérivées de la fonction du coût c jusqu'à l'ordre 4, dite le tenseur de Ma-Trudinger-Wang. Ils ont montré la régularité C 2 du potentiel sous la condition A3S, c'est-à-dire, le tenseur de MTW est strictement positive. Plus tard, Kim et McCann [START_REF] Kim | Continuity, curvature, and the general covariance of optimal transportation[END_REF] ont interprété de nouveau le tenseur de MTW comme la courbure de Riemann sur certains 2-plans d'une métrique pseudo-riemannienne issue de la fonction du coût sur l'espace produit M × M . A propos du tenseur de MTW ou plus généralement la courbure croisée, voir les références [START_REF] Loeper | Regularity of optimal transport in curved geometry: the nonfocal case[END_REF] [START_REF] Figalli | Nearly round spheres look convex[END_REF] [67] [START_REF] Figalli | On the Ma-Trudinger-Wang curvature on surfaces[END_REF] [73] [START_REF] Delanoë | Locally nearly spherical surfaces are almost-positively curved[END_REF] [START_REF] Delanoë | Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds[END_REF].

Avant de définir le tenseur de MTW, on donne quelques notations. Soit (M, g) une variété riemannienne compacte de dimension n ≥ 2. (exp m (tξ), exp m (ν + sη)).

Cette définition a un sens. En fait, lorsque t et s sont suffisamment petits, exp m (ν + sη) / ∈ Cut(exp m (tξ)). Et donc d 2 (exp m (tξ), exp m (ν + sη)) est lisse par rapport à t et s. On énonce quelques propriétés élémentaires: quand ξ = 0 ou η = 0, le tenseur de MTW s'annule. En général, si le rang de la famille de vecteurs {ν, ξ, η} est plus petit que 1, alors le tenseur de MTW s'annule. Il est clair que le tenseur de MTW est homogène de degré 2 par rapport à ξ ou η, et homogène de degré 1 par rapport à la métrique g, c'est-à-dire η). Lorsque (M, g) est plate, le tenseur de MTW s'annule.

C (m,ν) (λξ, η) = λ 2 C (m,ν) (ξ, η), C (m,ν) (ξ, λη) = λ 2 C (m,ν) (ξ, η); C λg (m,ν) (ξ, η) = λC g (m,ν) (ξ,
Le tenseur de MTW a des liens étroits avec la courbure de Riemann de la variété. Loeper [START_REF] Loeper | On the regularity of solutions of optimal transportation problems[END_REF] a trouvé que le tenseur de MTW sur le diagonal coïncide avec la courbure sectionnelle. En effet, on a De plus, le tenseur de MTW a un développement (voir [START_REF] Lee | New computable necessary conditions for the regularity theory of optimal transportation[END_REF])

d 2 (exp m tξ, exp m sη) = |ξ| 2 m t 2 -2g m (ξ, η)ts + |η| 2 m s 2 - 1 3 R m (ξ, η, ξ, η)t 2 s 2 + o((t 2 + s 2 ) 2 ).
C(m, ν)(ξ, η) = R m (ξ, η, ξ, η) + 1 2 (∇ η R) m (ξ, ν, ξ, η) + 1 4 (∇ ν R) m (ξ, η, ξ, η) + o(|ν| m ).

La condition de Ma-Trudinger-Wang

La régularité de l'application du transport optimal est liée à la positivité du tenseur de MTW. On introduit des conditions de courbure suivantes. Il est intéressant de trouver des variétés riemanniennes qui satisfont la condition de MTW. On donne quelques exemples. Lorsque M est plate (par exemple R n , T n ), la condition A3W est satisfaite, mais la condition A3S n'est pas satisfaite. Loeper [START_REF] Loeper | Regularity of optimal maps on the sphere: The quadratic cost and the reflector antenna[END_REF] a prouvé que la condition A3S est satisfaite sur la sphère S n . Kim-McCann [START_REF] Kim | Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular)[END_REF] ont montré que la submersion riemannienne de la sphère S n (par exemple CP n , HP n ) satisfait la condition A3S. Delanoë-Rouvière [START_REF] Delanoë | Positively curved Riemannian locally symmetric spaces are positively squared distance curved[END_REF] ont montré que la variété riemannienne symétrique à courbure sectionnelle strictement positive satisfait la condition A3S. Figalli et Rifford [START_REF] Figalli | Continuity of optimal transport maps on small deformations of S2[END_REF] ont prouvé que la condition A3S est vérifiée sur une surface simplement connexe dont la métrique est une perturbation de classe C 4 par rapport à celle de la sphère S 2 . Lorsque la courbure de Gauss de la surface s'approche de 1 en norme C 2 , Delanoë-Ge [START_REF] Delanoë | Locally nearly spherical surfaces are almost-positively curved[END_REF] ont obtenu la condition A3S là-dessus. Figalli-Rifford-Villani [START_REF] Figalli | Nearly round spheres look convex[END_REF] ont démontré que la condition A3S est satisfaite sur une variétés riemannienne compacte de dimension n dont la métrique est une perturbation de classe C 4 par rapport à la sphère canonique S n . Du-Li [START_REF] Du | Positivity of Ma-Trudinger-Wang curvature on Riemannian surfaces[END_REF] ont donné une condition suffisante pour que la condition A3S soit vérifiée sur une surface fermée.

On donne quelques remarques sur la condition A3W et la condition A3S. Tout d'abord, il est clair que la condition A3S implique la condition A3W, mais la réciproque est fausse en général.

Et puis, la condition A3W implique la positivité de la courbure sectionnelle (voir [START_REF] Loeper | On the regularity of solutions of optimal transportation problems[END_REF]). En revanche, la réciproque n'est pas vraie(voir [START_REF] Kim | Counterexamples to continuity of optimal transportation on positively curved Riemannian manifolds[END_REF]). De même, la condition A3S implique que la courbure sectionnelle est strictement positive.

Pour les autres fonctions du coût, la condition A3S n'implique pas forcément que la courbure sectionnelle est positive. Par exemple, on considère la fonction du coût c(•, •) =cosh d(•, •) sur l'espace hyperbolique H n . La condition A3S est satisfaite, mais la courbure sectionnelle de H n est toujours égale à -1.

En outre, la condition A3W et la condition A3S sont préservées pour une submersion de Riemann [START_REF] Kim | Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular)[END_REF]. La condition A3S est également stable sous la limite de Gromov-Hausdorff [START_REF] Villani | Stability of a 4th-order curvature condition arising in optimal transport theory[END_REF]. xiv CONTENTS On a d'autres caractérisations pour les conditions A3S et A3W. Loeper [START_REF] Loeper | On the regularity of solutions of optimal transportation problems[END_REF] a prouvé que, la c-convexité des ensembles de contact, la condition A3W et la connexité de la csousdifférentiel du potentiel c-convexe sont toutes équivalentes. Si le lieu de coupure d'un point n'est pas un lieu conjugué, Loeper-Villani [START_REF] Loeper | Regularity of optimal transport in curved geometry: the nonfocal case[END_REF] ont prouvé que la condition A3S implique la convexité uniforme de les domaines d'injectivité. Figalli-Gallouët-Rifford [START_REF] Figalli | On the Convexity of Injectivity Domains on Nonfocal Manifolds[END_REF] ont montré que la condition A3W implique la convexité des domaines d'injectivité sous des hypothèses convenables. Cependant, ce problème n'est pas complètement résolu.

Les conditions A3W et A3S jouent un rôle important dans la théorie de la régularité de l'application du transport optimal. Loeper [START_REF] Loeper | On the regularity of solutions of optimal transportation problems[END_REF], Villani [START_REF] Villani | Optimal transport, old and new[END_REF], Figalli-Rifford-Villani [START_REF] Figalli | Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds[END_REF] ont prouvé que la condition A3W est nécessaire pour la continuité de l'application du transport optimal.

On considère la fonction du coût égale à la distance au carré. Dans l'espace euclidien, la régularité de l'application du transport optimal a été entièrement résolue. Dans ce cas là, l'équation (3) est équivalente à l'équation de Monge-Ampère classique. L'issue de la régularité est obtenue par Caffarelli [START_REF] Caffarelli | The regularity of mapping with a convex potential[END_REF][START_REF] Caffarelli | Boundary regularity of maps with convex potentials[END_REF][START_REF] Caffarelli | Boundary regularity of maps with convex potentials II[END_REF], Delanoë [START_REF] Delanoë | Classical solvability in dimension two of the second boundary value problem associated with the Monge-Ampère operator[END_REF] et Urbas [START_REF] Urbas | On the second boundary value problem for equations of Monge-Ampère type[END_REF]. Sur une variété riemannienne, Cordero-Erausquin [START_REF] Cordero-Erausquin | Sur le transport de mesures périodiques[END_REF] a prouvé que l'application du transport optimal sur un tore T n est lisse(voir également [START_REF] Delanoë | On the smoothness of the potential function in Riemannian optimal transport[END_REF]). Loeper [START_REF] Loeper | Regularity of optimal maps on the sphere: The quadratic cost and the reflector antenna[END_REF] a montré que l'application du transport optimal sur une sphère standard S n est lisse. Si le lieu de coupure ne rencontre pas le lieu conjugué et si M satisfait la condition A3S, Loeper et Villani [START_REF] Loeper | Regularity of optimal transport in curved geometry: the nonfocal case[END_REF] ont obtenu le résultat de régularité. Pour le produit des sphères standards, la régularité de l'application du transport optimal est montrée par Figalli-Kim-McCann [START_REF] Figalli | Regularity of optimal transport maps on multiple products of spheres[END_REF]. Delanoë et Ge ont étudié ce problème de régularité sur des variétés riemanniennes dont la courbure est proche de celle de la sphère standard S n en C 2 norme. Delanoë [START_REF] Delanoë | On the smoothness of the potential function in Riemannian optimal transport[END_REF] a prouvé que l'application du transport optimale est lisse sur la variété symétrique à courbure sectionnelle strictement positive, et sur une surface dont la courbure de gauss est proche de 1 en C 2 norme.

Résultats principaux

Avant d'énoncer les résultats, on rappelle quelques notations et définitions. Soit (M, g) une variété riemannienne compacte connexe lisse sans bord de la dimension n ≥ 2. On dit brièvement une variété riemannienne fermée. Soit K la courbure sectionnelle de (M, g). La courbure de Riemann de (M, g) est notée par Riem. Le carré de la superficie du parallélogramme engendré par deux vecteurs tangents ξ, η ∈ T m M est égal à |ξ ∧ η| 2 m = |ξ| 2 m |η| 2 mg m (ξ, η) 2 . Soient X, Y, Z, W des champs de vecteur lisses sur M . Le produit de Kulkarni-Nomizu T 1 T 2 de deux champs de 2-tenseurs symétriques T 1 et T 2 est défini par

T 1 T 2 (X, Y, Z, W ) = T 1 (X, Z)T 2 (Y, W ) + T 1 (Y, W )T 2 (X, Z) - T 1 (X, W )T 2 (Y, Z) -T 1 (Y, Z)T 2 (X, W ).
On suppose toujours que la courbure sectionnelle de (M, g) satisfait min Gr2(M ) K = 1.

(

) 5
et la courbure de Riemann satisfait

∥Riem - 1 2 g g∥ C 2 (M,g) < ε (6) Pour ν ̸ = 0, on considère S(m, ν, 1)(ξ) = ξ -(1 -|ν| m cot |ν| m )(ξ -g m (ξ, ν |ν| m ) ν |ν| m ) (7)
et on note C(m, ν)(ξ, η) le tenseur de MTW sur la sphère standard S n , c'est-à-dire,

C(m, ν)(ξ, η) = - 3 2 d 2 ds 2 | s=0 g m ( S(m, ν + sη, 1)(ξ), ξ). (8) 0.4. R ÉSULTATS PRINCIPAUX xv Il est clair que lim ν→0 S(m, ν, 1)(ξ) = |ξ| 2 m et que lim ν→0 C(m, ν)(ξ, η) = Rm (ξ, η, ξ, η) = |ξ| 2 m |η| 2 m -g m (ξ, η) 2 . ( 9
)
Les résultats principaux dans cette thèse sont inclus dans deux prépublications (voir [START_REF] Ge | Regularity of the optimal transport maps on the nearly spherical manifold[END_REF][START_REF] Ye | Regularity of the optimal transport map on Riemannian products of nearly spherical manifolds[END_REF]).

Le premier résultat consiste à la stabilité de positivité du tenseur de MTW sur une variété presque sphérique. Théorème 0.1. Soit (M, g) une variété riemannienne fermée de dimension n ≥ 2. Suppose que (M, g) satisfait [START_REF] Bonnard | Conjugate-cut loci and injectivity domains on two-spheres of revolution[END_REF]. Alors il existe deux constantes strictement positives ε 0 , κ 0 > 0 qui ne dépendent que de n, telles que si [START_REF] Bonnard | Convexity of injectivity domains on the ellipsoid of revolution: The oblate case[END_REF]

est vérifée avec ε ≤ ε 0 , alors pour tout m ∈ M, ν ∈ I(m) et pour tous vecteurs tangents ξ, η ∈ T m M, on a C(m, ν)(ξ, η) ≥ κ 0 (|ξ ∧ η| 2 m + |ξ| 2 m |η ∧ ν| 2 m + |ξ ∧ ν| 2 m |η| 2 m ). ( 10
)
Une conséquence immédiate du théorème est la suivante:

Corollaire 0.

m ∈ M, v ∈ I(m), |v| ≥ 3π 4 . 1) |S -1 (m, v, 1) -S-1 (m, v, 1)| ≤ Cε; 2) |∂ x S -1 (m, v, 1) -∂ x S-1 (m, v, 1)| ≤ Cε, |D v S -1 (m, v, 1) -D v S-1 (m, v, 1)| ≤ Cε; 3) |∂ 2 xx S -1 (m, v, 1) -∂ 2 xx S-1 (m, v, 1)| ≤ Cε, |∂ x D v S -1 (m, v, 1) -∂ x D v S-1 (m, v, 1)| ≤ Cε, |D 2 vv S -1 (m, v, 1) -D 2 vv S-1 (m, v, 1)| ≤ Cε.
A l'aide de la méthode de continuité, on prouve le résultat de la régularité de l'application du transport optimal. Théorème 0.2. Soit (M, g) une variété riemannienne fermée de dimension n ≥ 2. Suppose que (M, g) satisfait [START_REF] Bonnard | Conjugate-cut loci and injectivity domains on two-spheres of revolution[END_REF]. Alors il existe une constante strictement positive ε 0 > 0 qui ne dépend que de n, telle que, si

∥Riem - 1 2 g g∥ C 2 (M,g) < ε 0 , alors pour tout (k, α) ∈ N × (0, 1), avec k ≥ 2,

xvi

CONTENTS

De la même manière, on obtient la régularité de l'application du transport optimal sur une variété produit des variétés presque sphériques. Théorème 0.3. Soient M 1 et M 2 deux variétés riemanniennes fermées de dimension n 1 ≥ 2 et n 2 ≥ 2 respectivement. Suppose que ∀i, (M i , g i ) satisfait [START_REF] Bonnard | Conjugate-cut loci and injectivity domains on two-spheres of revolution[END_REF]. Il existe une constante strictement positive ε 0 > 0 qui ne dépend que de n i pour i = 1, 2, telle que, si

∥Riem i - 1 2 g i g i ∥ C 2 (Mi,gi) < ε 0 ,
alors pour tout (k, α) ∈ N × (0, 1), avec k ≥ 2, et pour toutes mesures de probabilité à densité strictement positive de classe C k,α sur M 1 × M 2 ρ 0 dvol et ρ 1 dvol , le potentiel du transport optimal envoyant ρ 0 dvol vers ρ 1 dvol est de classe C k+2,α .

Une conséquence directe est la suivante.

Corollaire 0.3. Sous les mêmes hypothèses du Théorème 0.3, toutes mesures de probabilité à densité strictement positive de classe

C ∞ sur M 1 × M 2 ρ 0 dvol et ρ 1 dvol, le potentiel du transport optimal envoyant ρ 0 dvol vers ρ 1 dvol est de classe C ∞ .
Dans [START_REF] Ye | Regularity of the optimal transport map on Riemannian products of nearly spherical manifolds[END_REF], on montre également que si la métrique sur M 1 × M 2 n'est pas sous forme de produit, alors l'application du transport optimal n'est pas forcément régulière même si la métrique est proche du produit des sphères en norme C 4 . Plus précisement, on a Théorème 0.4. On note g × le produit des métriques canoniques sur

S n1 × S n2 avec n 1 ≥ 2 et n 2 ≥ 2. Alors ∀ε > 0, il existe une métrique g sur S n1 × S n2 conforme à g × satisfaisant ∥g -g × ∥ C 4 < ε
telle que l'on puisse trouver des mesures de probabilité à densité C ∞ strictement positive sur S n1 × S n2 dont l'application du transport optimal correspondante n'est pas lisse.

Chapter 1

Preliminaries

Basic notations and conventions

In this section, some basic notations from Riemannian geometry will be stated. See [START_REF] Alías | Maximum Principles and Geometric Applications[END_REF] [2] [START_REF] Carmo | Riemannian Geometry[END_REF] [19] [START_REF] Cheeger | Comparison Theorems in Riemannian Geometry[END_REF] [80] as references on Riemannian geometry.

Let (M, g) be a complete connected smooth Riemannian manifold of dimension n ≥ 2. Let X, Y, Z, W be smooth vector fields on M. The (3,1)-type Riemann curvature tensor of the Riemaniann manifold (M, g) is defined by

R(X, Y )Z = ∇ X ∇ Y Z -∇ Y ∇ X Z -∇ [X,Y ] Z
where ∇ denotes the Levi-Civita connection of g.

We set Riem for the associated (4,0)-type Riemann curvature tensor

1 , i.e. Riem(X, Y, Z, W ) = ⟨R(Z, W )Y, X⟩.
Throughout the thesis, we adopt the Einstein summation convention over repeated indices.

In a local coordinate system{x The Riemannian metric induces norms on all the tensor bundles. Precisely, the squared norm of (r, s)-tensor field T in the coordinate system

x = (x 1 , • • • , x n ) is given by |T | 2 = g i1k1 • • • g isks g j1l1 • • • g jrlr T i1•••is j1•••jr T k1•••ks l1•••lr , where T i1•••is j1•••
jr are components of T in the coordinate system x. We will need notation for the second covariant derivative of a tensor field, we write

∇ 2 X,Y T := (∇ 2 T )(X, Y, • • • ). It is remarkable to note that ∇ 2 X,Y T = ∇ X (∇ Y T ) -∇ ∇X Y T.
Another fact will be used frequently is that the tensor g g is parallel, i.e. ∇(g g) = 0.

(1.1)

For later use, the (3,1)-form of 1 2 g g is denoted by R, i.e. R(X, Y )Z = ⟨Y, Z⟩X -⟨X, Z⟩Y.

Given a local coordinate system{x

1 , • • • , x n }, the components of R are given by R( ∂ ∂x j , ∂ ∂x k ) ∂ ∂x i = Rl ijk ∂ ∂x l , with Rl ijk = δ l j g ik -δ l k g ij . (1.2) 1 We use g(•, •) and ⟨•, •⟩ interchangeably.
1

Let K : Gr 2 (M ) → R be the sectional curvature defined on the Grassmann bundle of tangent 2-planes(to see [START_REF] Delanoë | On the smoothness of the potential function in Riemannian optimal transport[END_REF]). We suppose that (M, g) be closed manifold of dimension n ≥ 2 throughout the thesis unless otherwise specified. We also assume the sectional curvature of M satisfies min

Gr2(M ) K = 1. (1.3)
and the Riemann curvature tensor satisfies

∥Riem - 1 2 g g∥ C 2 (M,g) < ε, with ε small. (1.4)
If (M, g) satisfies (1.3) and (1.4), we say that (M, g) is nearly spherical.

It is readily to see that (1.3) and (1.4) hold on round sphere S n .

In addition, by (1.4), it follows that

∥Ric -(n -1)g∥ C 2 (M,g) < ε, ∥Scal -n(n -1)∥ C 2 (M,g) < ε.
In two dimension, up to a constant, the curvature assumption (1.4) is equivalent to

|K -1| C 2 (M,g) < ε.
While in n ≥ 3 dimension, up to a constant, the curvature assumption (1.4) deduces

∥Riem - Scal 2n(n -1) g g∥ C 2 (M,g) < ε.
Indeed, from the well-known decomposition of Riem we obtain the identity

|Riem - 1 2 g g| 2 = |Riem - Scal 2n(n -1) g g| 2 + 2 n(n -1) (Scal -n(n -1)) 2 .
Together with the parallel property (1.1) and the definition of Scalar curvature, the result is derived.

The assumption (1.3) and (1.4) contain some geometric information of (M, g). In view of Bonnet myers theorem [START_REF] Cheeger | Comparison Theorems in Riemannian Geometry[END_REF], the normalization (1.3) implies that (M, g) is compact and there is at least one conjugate point along every geodesic. The compactness infers that there is cut point along every geodesic [START_REF] Carmo | Riemannian Geometry[END_REF].

Let m ∈ M. For ∀ν ∈ T m M, |ν| m = 1. Set t C (m, ν) be the distance from point m to the cut point of m along the geodesic exp m (tν), i.e.

t C (m, ν) = sup{t ≥ 0; exp m (sν)| 0≤s≤t is a minimizing geodesic}.
The injectivity domain at m is denoted by I(m), i.e. The geometry of injectivity domain is complicated. But on some special manifolds they have nice geometric properties. For instance, the injectivity domain of round sphere S n is the open ball of radius π centered at the origin. If the Riemannian manifold is simply connected, complete and with non-positive curvature, from theorem of Hadamard [START_REF] Carmo | Riemannian Geometry[END_REF], we know that the injectivity domains are R n . Figalli-Rifford-Villani [START_REF] Figalli | Nearly round spheres look convex[END_REF] established that the injectivity domains are uniformly convex on the Riemannian manifold which is the C 4 metric perturbation of round sphere S n . Bonnard [START_REF] Bonnard | Convexity of injectivity domains on the ellipsoid of revolution: The oblate case[END_REF] proved that the injectivity domain of the ellipsoid {(x, y, z) ∈ R 3 :

I(m) = {tν; 0 ≤ t < t C (m, ν), ν ∈ T m M \{0}}. The focal time t F (m, ν) is defined by t F (m, ν) = inf{t ≥ 0; exp m (tν)is conjugate to m}.
x 2 + y 2 + z 2 µ 2 = 1}(µ ∈ (0, 1]) is convex if and only if µ ≥ 1 √ 3 .
The notation exp -1 m (y) stands for all the velocities ν ∈ T m M such that the geodesic exp m (sν)| 0≤s≤1 is minimizing and exp m ν = y. Under the curvature assumption (1.3), by the Bonnet-Myers theorem [START_REF] Cheeger | Comparison Theorems in Riemannian Geometry[END_REF], for each ν ∈ exp -1 m (y), the length of ν is not great than π. Moreover, if ν ∈ I(m), by Rauch comparison theorem, the length of ν is strictly less than π. We will use these facts frequently throughout the thesis.

For y ∈ M, we consider the functions

d 2 y 2 (•) = 1 2 d 2 (•, y) which is smooth in M \Cut y . For any m / ∈ Cut y , the Gauss lemma implies that grad d 2 y 2 (m) = -exp -1 m y. (1.5)
Given a real smooth function u defined on M. The Hessian of u at m is given by the linear operator from T m M to T m M defined by the identity

for ∀ξ ∈ T m M, ∇ 2 m u(ξ) := ∇ ξ (grad u).
It is easy to see that the Hessian is self-adjoint and can be calculated as follows

⟨∇ 2 m u(ξ), ξ⟩ = d 2 ds 2 s=0 u(γ(s)), (1.6)
where γ is a geodesic with the initial point m and the initial velocity ξ.

Jacobi matrix 1.2.1 Initial Jacobi matrices

In this section, we give the definition and some basic facts about the initial Jacobi matrices. The definition is stated as follows. See Chapter 14 in [START_REF] Villani | Optimal transport, old and new[END_REF] as references on the initial Jacobi matrices and the Jacobi matrix.

Definition 1.1. Given m ∈ M, ν ∈ T m M \{0}. Let {E 1 , E 2 , • • • , E n } be an orthonormal basis of T m M with E 1 = ν/|ν| m . Let γ(•) be a geodesic with initial point m and initial velocity ν and {e 1 , e 2 , • • • , e n } be the parallel transport of {E 1 , E 2 , • • • , E n } along γ with e i (0) = E i .
We define the matrices J 0 (m, ν, t) and J 1 (m, ν, t) as the matrix valued solutions of the second order equation

Ja + RJ a = 0, a = 0, 1, (1.7)
with the initial condition

J 0 (m, ν, 0) = 0, J0 (m, ν, 0) = I n , J 1 (m, ν, 0) = I n , J1 (m, ν, 0) = 0,
where the elements of R are given by R ij (t) = ⟨R(e i (t), γ(t)) γ(t), e j (t)⟩.

(1.8)

The matrices J 0 (m, ν, t) and J 1 (m, ν, t) are called the initial Jacobi matrices. The matrix (R ij ) is called curvature matrix. The equation (1.7) is called the Jacobi equation. A matrix J is called Jacobi matrix if it satisfies the Jacobi equation (1.7).

We give some facts about the curvature matrix. It is clear that the curvature matrix (R ij ) has vanishing first row and first column. The curvature matrix (R ij ) is symmetric and its the trace gives the Ricci curvature. In addition, the curvature matrix (R ij ) is positive semi-definite if the sectional curvature of M is non-negative.

As same as on the sphere, we define Ja (m, ν, t) as the matrix-valued of the second order equation

     Ja + R Ja = 0, a = 0, 1, J0 (m, ν, 0) = 0, J0 (m, ν, 0) = I n , J1 (m, ν, 0) = I n , J1 (m, ν, 0) = 0.
(1.9)

The elements of R are given by Rij (m, ν, t) = ⟨ R(e i (t), γ(t)) γ(t), e j (t)⟩.

It is readily to see that J0 and J1 are given respectively by J0 =

[ t 0 0 sin(|ν|t) |ν| I n-1 ] , J1 = [ 1 0 0 cos(|ν|t)I n-1 ]
.

From the homogeneity of a geodesic(to see [START_REF] Carmo | Riemannian Geometry[END_REF] p.64), we get the homogeneity of the initial Jacobi matrices,i.e.

λJ 0 (m, λν, t) = J 0 (m, ν, λt), J 1 (m, λν, t) = J 1 (m, ν, λt), λ > 0.
For t ∈ [0, 1], a = 0, 1, we can extend the initial matrix J a by continuity at ν = 0 by J a (m, 0, t) = I n . Similarly, we can extend J0 , J

(k) 0 , k ≥ 2 at ν = 0 by J0 (m, 0, t) = I n , J (k) 0 (m, 0, t) = 0 and J (k) 1 , k ≥ 1 at ν = 0 by J (k) 1 (m, 0, t) = 0.
For simplicity, the initial Jacobi matrix J a (m, ν, t) is abbreviated to J a (t) unless otherwise specified.

By definition of the conjugate point, the initial matrix J 0 (t) is invertible for every t ∈ (0, t F (m, ν)). Moreover, in view of Proposition 14.30 in [START_REF] Villani | Optimal transport, old and new[END_REF], the continuity derives that det J 0 (t) > 0 for every t ∈ (0, t F (m, ν)).

We now present the Hessian of the squared distance in terms of the initial Jacobi matrix and a representation formula of the inhomogeneous Jacobi equation. Proposition 1.1. Under the hypothesis of Definition 1.1, we have (a) Let J(t) be the Jacobi field along the geodesic exp m (tν) defined by the conditions

J(0) = ξ, J(1) = 0. Then J(t) = -J 0 (t)J -1 0 (1)J 1 (1)(ξ) + J 1 (t)(ξ); (b) For t ∈ [0, t F (m, ν)). Let S(m, ν, t) be the linear operator from T m M to T m M whose matrix in the orthonormal basis {E 1 , E 2 , • • • , E n } is given by tJ 0 (t) -1 J 1 (t). Then the linear operator S(m, ν, t) : T m M → T m M is self adjoint. Moreover, if ν ∈ I(m), then for ∀ξ ∈ T m M, ⟨∇ 2 m c(•, exp m ν)(ξ), ξ⟩ = ⟨S(m, ν, 1)(ξ), ξ⟩.
(c) (Representation formula)The solution of the matrix valued inhomogeneous Jacobi equation

J(t) + R(t)J(t) = B(t)
is given by the formula

J(t) = J 0 (t) J(0) + J 1 (t)J(0) + J 0 (t) ∫ t 0 J * 1 Bds -J 1 (t) ∫ t 0 J * 0 Bds,
where * means transpose of matrix.

Remark 1.1. 1.(Homogeneity)From the homogeneity of the initial Jacobi matrices, we infer that S(m, λν, t) = S(m, ν, λt), λ > 0.

Thus we can extended S by continuity at ν = 0 by S(m, 0, t) = Id; 2.The linear operator S(m, ν, t) has explicit formula on space forms [START_REF] Paul | New examples satisfying Ma-Trudinger-Wang conditions[END_REF], for instance, on the round sphere S n ,

S(m, ν, t)(ξ) = ξ -(1 -t|ν| cot(t|ν|))(ξ -⟨ξ, ν |ν| ⟩ ν |ν| ).
Equivalently, the associated covariant symmetric 2-tensor field is given by g

-(1 - t|ν| cot(t|ν|))(g -ν |ν| ⊗ ν |ν| ). 3.
We obtain the representation formula for the scalar function,i.e.

f (t) = f (0) cos t + ḟ (0) sin t + sin t ∫ t 0 ϕ(s) cos sds -cos t ∫ t 0 ϕ(s) sin sds, ( 1.10
)

where ϕ = f + f.
Proof. (a) is direct result of the uniqueness of the second order ordinary differential equation.

(b)The self-adjoint property of S refers to Proposition 14.30 in [START_REF] Villani | Optimal transport, old and new[END_REF]. We prove the second assertion here. Let ξ be a tangent vector based at m. If ν = 0, tJ 0 (t) -1 can be extended by continuity at t = 0 by I n . So

⟨∇ 2 c(•, m)(ξ), ξ⟩ = |ξ| 2 = ⟨S(m, ν, 1)(ξ), ξ⟩. If ν ∈ I(m)\{0}, then the curve σ(t) = exp m (tν)| t∈[0,1]
is the unique minimizing geodesic from m to exp m ν.

Let γ(•) be the geodesic with the initial point m and the initial velocity ξ. Consider the family of the geodesics σ(t, s) = exp γ(s) (t exp -1 γ(s) (exp m ν)), so that σ(t, 0) = σ(t). By the definition of the Jacobi field, it follows that J(t) = ∂ ∂s | s=0 σ is a Jacobi field along the geodesic σ(t) with J(0) = ξ and J(1) = 0. Moreover,

J(0) = ∂ 2 ∂s∂t | s=t=0 σ = d ds | s=0 exp -1 γ(s) (exp m ν) = -∇ 2 m c(•, exp m ν)(ξ).
where the last equality follows from (1.5) and the definition of Hessian.

Hence ⟨∇ 2 m c(•, exp m ν)(ξ), ξ⟩ = -⟨ J(0), J(0)⟩. From (a) the term J(t) is equal to -J 0 (t)J -1 0 (1)J 1 (1)(ξ) + J 1 (t)(ξ). So J(0) = -S(m, ν, 1)(ξ). Therefore ⟨∇ 2 m c(•, exp m ν)(ξ), ξ⟩ = ⟨S(m, ν, 1)(ξ), ξ⟩. (c)
The result follows from a direct calculation(the details refer to Lemma 3.2 in [START_REF] Figalli | Nearly round spheres look convex[END_REF]). This finishes the proof of Proposition 1.1.

The approximation of initial Jacobi matrices

In this subsection, we present the approximation of the initial Jacobi matrices. We first give a basic fact from the theory of second order differential equations [START_REF] Delanoë | Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds[END_REF].

Lemma 1.1. Let (M, g) be a closed manifold of dimension n ≥ 2. Suppose that M satisfies the curvature assumptions (1.3) and (1.4). Then there exists a positive constant C 1 depending only on n such that for every m ∈ M, ν ∈ T m M and every t ∈ [0, 1], the g norms of the initial Jacobi matrices: Keeping in mind the assumption (1.4), the initial Jacobi matrices can be estimated as follows [START_REF] Delanoë | Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds[END_REF]. Lemma 1.2. Let (M, g) be a closed manifold of dimension n ≥ 2. Suppose that M satisfies the curvature assumptions (1.3) and (1.4). Then there exists a positive constant C 2 depending only on n such that for every m ∈ M, ν ∈ T m M and every t ∈ [0, 1], the following estimates hold: Proof. It is obvious for ν = 0. Without generality, we assume that ν ̸ = 0. The length of the tangent vector ν is denoted by τ.

|J a (m, ν, t)|, | Ja (m, ν, t)|, a = 0, 1.
|J a (m, ν, t) -Ja (m, ν, t)| ≤ C 2 ε, | Ja (m, ν, t) -Ja (m, ν, t)| ≤ C 2 ε, a = 0, 1.
Note that the Jacobi equation (1.7) can be rewritten as follows:

Ja + RJ a = ( R -R)J a .
Applying a representation formula in [START_REF] Delanoë | Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds[END_REF] to the elements of the matrix J a , we have

J a = Ja + τ 2 sin(τ t) ∫ t 0 ds sin 2 (τ s) ∫ s 0 sin(τ θ)( R -R)J a dθ.
Using (1.3) and the Cauchy-Schwarz inequality, we derive

|J a -Ja | ≤ ε max t∈[0,1] |J ⊥ a |τ 2 sin(τ t) ∫ t 0 ds sin 2 (|ν 0 |s) ∫ s 0 sin(τ θ)dθ .
Together with Lemma 1.1 and the expression 1cos(τ t) = τ 2 sin(τ t)

∫ t 0 ds sin 2 (τ s) ∫ s 0 sin(τ θ)dθ,
we obtain that there exists a positive constant C depending only n such that |J a (m, ν, t) -Ja (m, ν, t)| ≤ Cε, a = 0, 1.

(1.11)

Note that Remark 1.2 follows from the Rauch comparison theorem [START_REF] Carmo | Riemannian Geometry[END_REF]. For the second inequality, from the initial condition Ja (0) = Ja (0), we get Ja -Ja =

∫ t 0 ( Ja -Ja ) ds = ∫ t 0 ( R Ja -RJ a ) ds = ∫ t 0 R( Ja -J a ) + ( R -R)J a ds.
Together with the curvature assumption (1.4), Lemma 1.1 and (1.11), we imply that there exist a positive constant C such that

| Ja (m, ν, t) -Ja (m, ν, t)| ≤ Cε.
This finishes the proof of Lemma 1.2 by choosing C 2 = max{C, C}.

Fermi coordinate system

In this subsection, we state the definition of Fermi coordinate system. Definition 1.2. (Fermi coordinate system) Let (M, g) be an n-dimensional Riemannian manifold. Given a compact interval I ⊂ R which contains 0. Let γ : I → M be a geodesic with | γ| = 1 and {e 1 (t), e 2 (t), • • • , e n (t)} be a parallel orthonormal moving frame of vector fields along the geodesic γ with e 1 (t) = γ(t). The Fermi coordinate system {x 1 , • • • , x n } are defined by

x 1 (exp γ(t) ( n ∑ β=2 λ β e β (t))) = t, x α (exp γ(t) ( n ∑ β=2 λ β e β (t))) = λ α , 2 ≤ α ≤ n, t ∈ I.
where λ β are sufficiently small so that the exponential maps are defined.

In the Fermi coordinate system {x 1 , • • • , x n }, γ is called the axis.

Observe that the differential of the map (

x 1 , • • • , x n ) → exp γ(x 1 ) ( ∑ n β=2
x β e β (t)) on the axis is equal to the identity. Thanks to the inverse function theorem, the Fermi coordinate system makes sense.

The Fermi coordinate system is generalization of the normal coordinate system. To see this, along the axis we have

∀i, j, k ∈ {1, 2, • • • , n}, g ij (x 1 , 0) = δ ij , ∂ k g ij (x 1 , 0) = 0.
(1.12)

We will require higher order derivatives of the metric and Christoffel symbols on the axis. In the following, the Latin indices run over 1, • • • , n and the Greek indices run over 2, • • • , n.

On the axis, we have the following expressions.

Lemma 1.3. The following identities hold on the axis:

∂ 2 ij g 11 = -2R 1i1j , ∂ 2 αβ g 1µ = - 2 3 (R α1βµ + R αµβ1 ), (1.13) ∂ 2 αβ g ρµ = - 1 3 (R αρβµ + R αµβρ ), (1.14)
∂ k Γ i 1j = R i jk1 , ∂ α Γ i βµ = 1 3 (R i βαµ + R i µαβ ), (1.15)
∂ 2 αβ Γ i 11 = ∇ 1 R i βα1 + ∇ α R i 1β1 , (1.16) ∂ 2 αβ Γ 1 1µ = 1 3 (∇ 1 R 1 βαµ -∇ 1 R 1 µβα ) -∇ α R 1 µ1β , (1.17) ∂ 2 αβ Γ ρ 1µ = 1 2 (∇ α R ρ µβ1 + ∇ β R ρ µα1 ) + 1 6 (∇ 1 R ρ αβµ + ∇ 1 R ρ βαµ ). (1.18)
Furthermore, applying p times ∂ ∂x 1 (axis-derivative) to any of the preceding left side quantities, yields on the axis the p-th covariant derivative ∇ p 1 of the corresponding intrinsic right side quantity. For instance:

∂ 1 (∂ k Γ i 1j ) = ∇ 1 R i jk1 .
Proof. The results are given in [START_REF] Delanoë | Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds[END_REF] except (1.14). We only prove (1.14) here. By definition of the Riemann curvature tensor, on the axis, we have

R αρβµ = 1 2 (∂ 2 βρ g αµ + ∂ 2 αµ g βρ -∂ 2 αβ g ρµ -∂ 2 ρµ g αβ ).
Thanks to the identity:

∂ 2 ρµ g αβ = ∂ 2 ρµ g αβ , Chapter 1. Preliminaries
We derive

R αρβµ = ∂ 2 αµ g βρ -∂ 2 αβ g ρµ . Thus R αρβµ + R αµβρ = ∂ 2 αµ g βρ + ∂ 2 αρ g βµ -2∂ 2 αβ g ρµ = -3∂ 2 αβ g ρµ .
where the last inequality follows the well known identity

∂ 2 µα g βρ + ∂ 2 µβ g αρ + ∂ 2 µρ g βα = 0.
This finishes the proof of Lemma 1.3.

Chapter 2

The behaviour of the curvature matrix

In this chapter, we study the behaviour of the curvature matrix (R ij ). We start by giving some notations. Let (M, g) be an n-dimensional Riemannian manifold satisfying (1.3) and (1.4) for some small positive number ε. Fix m 0 ∈ M and ν 0 ∈ I(m 0 )\{0}. The length of the tangent vector ν 0 is denoted by τ.

Taking the orthonormal basis

{E 1 , E 2 , • • • , E n } of T m0 M.
Let the curve γ θ (t) be the geodesic with the initial point m 0 and the initial velocity cos θE 1 + sin θE 2 . For |θ| sufficiently small, let {e 1 (θ, t), e 2 (θ, t), • • • , e n (θ, t)} be the parallel transport along the geodesic γ θ (t) with e 1 (θ, 0) = cos θE 1 + sin θE 2 , e 2 (θ, 0) =sin θE 1 + cos θE 2 , e i (θ, 0) = E i for i ≥ 3. Then {e 1 (0, t), e 2 (0, t), • • • , e n (0, t)} is the parallel orthonormal moving frame along the geodesic γ 0 (t) with e 1 (0, t) = γ0 (t).

Let

x = (x 1 , x 2 , • • • , x n ) be the Fermi coordinate system along the geodesic γ 0 (t) and v = (v 1 , v 2 , • • • , v n )
be the fiber coordinates of T M → M naturally associated to x. We abbreviate the partial derivatives as follows:

∂ i = ∂ ∂x i , ∂ 2 ij = ∂ 2 ∂x i ∂x j , D i = ∂ ∂v i , D 2 ij = ∂ 2 ∂v i ∂v j .
For each m ∈ M, ν ∈ I(m) with m in the domain of the Fermi coordinate system x, we set:

X = X(x, v, t) = (X 1 (x, v, t), X 2 (x, v, t), • • • , X n (x, v, t)) = x(exp m (tν)), where x = x(m) and ν = v i ∂ i .
As t → exp m (tν) is a geodesic, thus the n-tuple X = X(x, v, t) is the solution of the following Cauchy problem:

{ Ẍi + Γ i jk (X) Ẋj Ẋk = 0, X i (x, v, 0) = x i , Ẋi (x, v, 0) = v i .
(2.1)

In the sequel, the dot will stand for the derivative with respect to t and the prime for the derivative with respect to θ. We will say that a constant is under control whenever it depends only on the dimension n. Given two real function f (t) and h(t), we write f (t) = B(h(t)) if there exists a positive constant C under control such that |f (t)| ≤ C|h(t)| for all t in a given range. The third derivative of f (t) with respect to t will be denoted by f (t).

The geodesic motion

In this subsection, we examine the geodesic motion on the axis. Let X(θ, t) = (X 1 (θ, t), X 2 (θ, t), • • • , X n (θ, t)) denotes the coordinate of the geodesic γ θ (t) in the 9 Fermi coordinate system x, i.e.

X i (θ, t) = X i (0, (cos θ, sin θ, 0, • • • , 0), t).
The geodesic motion is stated as follows.

Lemma 2.1. Let (M, g) be a closed Riemannian manifold of dimension n ≥ 2. Assume that (M, g) satisfies (1.3) and (1.4). Fix m 0 ∈ M and ν 0 ∈ I(m 0 )\{0}. In the Fermi coordinate system x, for any t ∈ [0, τ ], the geodesic motion X on the axis satisfies:

1) X ′ (0, t) = (0, sin t + B(εt 3 ), B(εt 3 ), • • • , B(εt 3 )) T , Ẋ′ (0, t) = (0, cos t + B(εt 2 ), B(εt 2 ), • • • , B(εt 2 )) T , Ẍ′ (0, t) = (0, -sin t + B(εt), B(εt), • • • , B(εt)) T ; 2) X ′′ (0, t) = (-sin t cos t + B(εt 3 ), B(εt 3 ), • • • , B(εt 3 )) T , Ẋ′′ (0, t) = (-cos(2t) + B(εt 2 ), B(εt 2 ), • • • , B(εt 2 )) T , Ẍ′′ (0, t) = (4 sin t cos t + B(εt), B(εt), • • • , B(εt)) T ; 3) Ẍ′ (0, 0) = (0, -R 2 121 (0), • • • , -R n 121 (0)) T , Ẍ′′ (0, 0) = (-4R 1 221 (0), 0, -4R 3 221 (0), • • • , -4R n 221 (0)) T .
Proof. Under the above assumptions, from (2.1), we know that X(θ, t) is the solution of the following Cauchy problem:

{ Ẍi + Γ i jk (X) Ẋj Ẋk = 0, X(θ, 0) = 0, Ẋ(θ, 0) = (cos θ, sin θ, 0, • • • , 0) T . (2.2)
On the axis, since the Christoffel symbols vanish, we have

X(0, t) = (t, 0, • • • , 0) T . ( 2.3)
We will settle Lemma 2.1 from 1) to 3) term by term. 1). We first handle the term Ẍ′ . Differentiating (2.2) with respect to θ, it follows that

Ẍi ′ + ∂ p Γ i jk (X) Ẋj Ẋk X p ′ + 2Γ i jk (X) Ẋj ′ Ẋk = 0,
with the initial condition

X i ′ (0, 0) = 0, Ẋi ′ (0, 0) = δ i 2 .
Evaluated on the axis, from (1.15),(2.3) and (1.12), we get the following equations

{ Ẍi ′ + R i 1α1 (X)X α ′ = 0, X ′ (0, 0) = 0, Ẋ′ (0, 0) = (0, 1, 0, • • • , 0) T . (2.4) It is clear that X 1 ′ (0, t) ≡ 0.
For i > 1, we first establish the following standard estimate.

Claim 2.1.

For any t ∈ [0, τ ] ⊂ [0, π], max{|X ′ |, | Ẋ′ |} ≤ e π 2 .
Proof of Claim 2.1.

Let f 1 = |X ′ | 2 + | Ẋ′ | 2 with f 1 (0) = 1. The derivative of f 1 takes the form ḟ1 = 2( Rβ 1α1 (X) -R β 1α1 (X))X α ′ Ẋβ ′ ,
where the term Rβ 1α1 is defined by (1.2). Suppose that ε ≤ 1, using the Cauchy-Schwarz inequality, we derive that ḟ1 ≤ f 1 .

Finally f 1 ≤ e t ≤ e π .
This completes the proof of Claim 2.1. Before proceeding, we define X

′ (t) = (X 1 ′ (t), • • • , X n ′ (t)) as follows { Ẍi ′ + Ri 1α1 (X)X α ′ = 0, X ′ (0, 0) = 0, Ẋ′ (0, 0) = (0, 1, 0, • • • , 0) T . Let E i be the difference X i ′ -X i ′ .
Let us rewrite the equation (2.4) in the perturbative form:

Ëi + E i = ( Ri 1α1 -R i 1α1 )X α ′ ,
with the null initial conditions:

E i (0) = Ėi (0) = 0.
Applying the representation formula of scalar function (1.10) to E i , it follows that

E i = sin t ∫ t 0 ( Ri 1α1 -R i 1α1 )X α ′ cos sds -cos t ∫ t 0 ( Ri 1α1 -R i 1α1 )X α ′ sin sds.
From the inequality sin t ≤ t in [0, τ ), the Cauchy-Schwarz inequality and Claim 2.1, we infer that

|E i | ≤ 3 2 e π 2 εt 2 . Precisely, |X 2 ′ -sin t| ≤ 3 2 εt 2 e π 2 ≤ 3 2 t 2 e π 2 , |X j ′ | ≤ 3 2 εt 2 e π 2 ≤ 3 2 t 2 e π 2 , 3 ≤ j ≤ n. Keep in mind that Ëi = -E i + ( Ri 1α1 -R i 1α1 )X α ′
, by the Cauchy-Schwarz inequality and the inequality sin t ≤ t in [0, τ ) again, we conclude

| Ëi | ≤ |E i | + ε|X ′ | ≤ ( √ 9 4
π 2 e π (n -1) + 3πe

π 2 + 1 + 3 2 πe π 2 ) εt,
as desired.

By integrating E i with respect to t, we derive the expression for X ′ and Ẋ′ ,i.e.

Ė(t) = ∫ t 0 Ë(s)ds = B(εt 2 ), E(t) = ∫ t 0 Ė(s)ds = B(εt 3 ).
2). As similar as 1), we first deal with Ẍ′′ . Differentiating the equation (2.2) twice with respect to θ :

Ẍi ′′ + ∂ p Γ i jk Ẋj Ẋk X p ′′ + ∂ 2 pq Γ i jk Ẋj Ẋk X p ′ X q ′ + 4∂ p Γ i jk Ẋj ′ Ẋk X p ′ + 2Γ i jk ( Ẋj ′′ Ẋk + Ẋj ′ Ẋk ′ ) = 0,
with the initial condition

X i ′′ (0, 0) = 0, Ẋi ′′ (0, 0) = -δ i 1 .
Evaluating on the axis, from (1.15),(2.3), (1.16) and (1.12), we get the following equations

{ Ẍi ′′ + R i 1α1 X α ′′ + (∇ α R i 1β1 + ∇ 1 R i βα1 )X α ′ X β ′ + 4R i βα1 X α ′ Ẋβ ′ = 0, X ′′ (0, 0) = 0, Ẋ′′ (0, 0) = (-1, 0, • • • , 0) T .
Similarly, we need the following Claim.

Claim 2.2.

There exists a positive constant C under control such that, for any t ∈

[0, τ ] ⊂ [0, π], max{|X ′′ |, | Ẋ′′ |} ≤ 2e π 2 C . Proof of Claim 2.2. Let f 2 (t) = |X ′′ | 2 + | Ẋ′′ | 2 . Then ḟ2 (t) = 2X i ′′ Ẋi ′′ + 2 Ẋi ′′ Ẍi ′′ = 2X i ′′ Ẋi ′′ -2R i 1α1 X α ′′ Ẋi ′′ -2[(∇ α R i 1β1 + ∇ 1 R i βα1 )X α ′ X β ′ + 4R i βα1 X α ′ Ẋβ ′ ] Ẋi ′′ ≤ Cf 2 (t) + C. We thus conclude that f 2 (t) ≤ 2e Ct ≤ 2e Cπ .
This ends the proof of Claim 2.2. We go back to the proof of Lemma 2.1.

If i = 1, let f 3 = X 1 ′′ + sin t cos t, then f3 = -∇ 1 R 1 βα1 X α ′ X β ′ + 4( R1 βα1 -R 1 βα1 )X α ′ Ẋβ ′ - 4 R1 βα1 X α ′ Ẋβ ′ -4 sin t cos t = 4X α ′ Ẋα ′ -4 sin t cos t + B(εt) = 4X 2 ′ Ẋ2 ′ -4 sin t cos t + B(εt) = 4(X 2 ′ -sin t) Ẋ2 ′ + 4( Ẋ2 ′ -cos t) sin t + B(εt) = B(εt).
If i > 1, the term X i ′′ satisfies the following equation

Ẍi ′′ + X i ′′ = ( Ri 1α1 -R i 1α1 )X α ′′ -(∇ α R i 1β1 + ∇ 1 R i βα1 )X α ′ X β ′ -4R i βα1 X α ′ Ẋβ ′ ,
with the homogenuous initial condition

X i ′′ (0, 0) = Ẋi ′′ (0, 0) = 0. (2.5)
Using the representation formula of scalar function (1.10) to X i ′′ , we see that

X i ′′ = sin t ∫ t 0 [( Ri 1α1 -R i 1α1 )X α ′′ -(∇ α R i 1β1 + ∇ 1 R i βα1 )X α ′ X β ′ - 4R i βα1 X α ′ Ẋβ ′ ] cos sds -cos t ∫ t 0 [( Ri 1α1 -R i 1α1 )X α ′′ - (∇ α R i 1β1 + ∇ 1 R i βα1 )X α ′ X β ′ -4R i βα1 X α ′ Ẋβ ′ ] sin sds. By Ri 1α1 -R i 1α1 = B(ε), ∇ α R i 1β1 = B(ε), ∇ 1 R i βα1 = B(ε), R i βα1 = B(ε), Claim 2.
1 and Claim 2.2, we have

X i ′′ = B(εt).
Arguing as above, we have

Ẍ′′ (0, t) -(4 sin t cos t, 0, • • • , 0) T = B(εt).
By integrating, together with the initial condition (2.5), we obtain

Ẋ′′ (0, t) -(-cos(2t), 0, • • • , 0) T = B(εt 2 ), X ′′ (0, t) -(-sin t cos t, 0, • • • , 0) T = B(εt 3 ).
3). The results are derived by tedious computation. We include the details in the following. Differentiating (2.2) once with respect to θ and t respectively:

Ẍi ′ + ∂ 2 pq Γ i jk Ẋj Ẋk Ẋp X q ′ + ∂ p Γ i jk ( Ẍj Ẋk X p ′ + Ẋj Ẍk X p ′ + Ẋj Ẋk Ẋp ′ ) + 2∂ p Γ i jk Ẋj ′ Ẋk Ẋp + 2Γ i jk ( Ẍj ′ Ẋk + Ẋj ′ Ẍk ) = 0.
Evaluating at the origin, and combining with the following relations X ′ (0, 0) = 0, Ẋ(0, 0) = (1, 0, • • • , 0) T , Ẋ′ (0, 0) = (0, 1, 0, • • • , 0) T , (1.15) and ∂ 1 Γ i jk (0) = Γ i jk (0) = 0, we conclude Ẍi ′ (0, 0) = -R i 121 (0). Differentiating the equation (2.2) twice with respect to θ and once with respect to t respectively, we have

Ẍi ′′ + ∂ 2 pq Γ i jk Ẋj Ẋk Ẋp X q ′′ + ∂ p Γ i jk (2 Ẍj Ẋk X p ′′ + Ẋj Ẋk Ẋp ′′ ) + ∂ 3 pql Γ i jk Ẋj Ẋk X p ′ X q ′ Ẋl + 2∂ 2 pq Γ i jk ( Ẍj Ẋk X p ′ X q ′ + Ẋj Ẋk Ẋp ′ X q ′ ) + 4∂ 2 pq Γ i jk Ẋj ′ Ẋk Ẋp X q ′ + 4∂ p Γ i jk ( Ẍj ′ Ẋk X p ′ + Ẋj ′ Ẍk X p ′ + Ẋj ′ Ẋk Ẋp ′ ) + 2∂ p Γ i jk ( Ẋj ′′ Ẋk + Ẋj ′ Ẋk ′ ) Ẋp + 2Γ i jk ( Ẍj ′′ Ẋk + Ẋj ′′ Ẍk + 2 Ẍj ′ Ẋk ′ ) = 0.
Evaluating at the origin, and combining with the following relations

X ′′ (0, 0) = 0, Ẋ′′ (0, 0) = (-1, 0 • • • , 0) T , ∂ 1 Γ i jk (0) = 0, X ′ (0, 0) = 0, Ẋ′ (0, 0) = (0, 1, 0 • • • , 0) T , Ẋ(0, 0) = (1, 0, • • • , 0) T , (1.15) and Γ i jk (0) = 0, we have Ẍi ′ (0, 0) = -4R i 221 (0).
The Lemma 2.1 is proved.

The orthonormal chart motion

To proceed, we study the orthonormal chart motion.

Along the geodesic γ θ (t), there are two charts:the natural chart { ∂ ∂x i } and the orthonormal chart {e 1 , • • • , e 2 }. To differentiate the curvature matrix (1.8), we need the coordinate of {e

1 , • • • , e 2 }. Set (Y j i (θ, t)) for the coordinates of the orthonormal chart {e 1 , • • • , e 2 }, i.e. e i (θ, t) = Y j i (θ, t)∂ j . It is clear that Y i 1 (θ, t) = Ẋi (θ, t). Since the orthonormal moving chart {e 1 , • • • , e n }
is parallel, we have the equation:

Ẏ i j + Γ i kl (X) Ẋl Y k j = 0, (2.6)
with the initial condition

Y (θ, 0) =   cos θ -sin θ sin θ cos θ I n-2   .
It is useful to note that Y i 1 = Ẋi . Moreover, it is obvious that Y (0, t) = I n . The orthonormal chart motion is presented as follows.

Lemma 2.2. Let (M, g) be a closed Riemannian manifold of dimension n ≥ 2. Assume that (M, g) satisfies (1.3) and (1.4). Fix m 0 ∈ M and ν 0 ∈ I(m 0 )\{0}. In the Fermi coordinate system x, for any t ∈ [0, τ ], the derivatives of the orthonormal chart motion along the axis satisfy:

1) Y i ′ j (0, t) = -(δ i 1 δ 2 j -δ i 2 δ 1 j ) cos t + B(εt 2 ), Ẏ i ′ j (0, t) = (δ i 1 δ 2 j -δ i 2 δ 1 j ) sin t + B(εt); 2) Y i ′′ j (0, t) = -δ i 1 δ 1 j cos(2t) -δ i 2 δ 2 j cos 2 t+ 1 3 (δ i j -δ i 1 δ 1 j -δ i 2 δ 2 j ) sin 2 t + B(εt 2 ), Ẏ i ′′ j (0, t) = 4δ i 1 δ 1 j sin t cos t + 2δ i 2 δ 2 j sin t cos t+ 2 3 (δ i j -δ i 1 δ 1 j -δ i 2 δ 2 j ) sin t cos t + B(εt); 3) Ÿ i ′ j (0, 0) = -R i j21 (0), Ÿ i ′′ j (0, 0) = 2δ 2 j R i 121 (0) -2 3 (1 + 5δ 1 j )R i 22j (0).
Proof. We will settle Lemma 2.2 from 1) to 3) term by term. 1). Similar to the proof of Lemma 2.1, we first consider Ẏ ′ . Differentiating the equation (2.6) with respect to θ :

Ẏ i ′ j + ∂ p Γ i kl Ẋl X p ′ Y k j + Γ i kl ( Ẋl ′ Y k j + Ẋl Y k ′ j ) = 0,
Evaluating on the axis, and replacing Ẋ, Y, Γ i kl , ∂ p Γ i k1 by the related values, we get

Ẏ i ′ j + R i jα1 X α ′ = 0,
with initial condition

Y i ′ j (0, 0) = δ i 2 δ 1 j -δ i 1 δ 2 j . ( 2.7)
Then

Ẏ i ′ j (0, t) = -R i jα1 X α ′ = ( Ri jα1 -R i jα1 )X α ′ -Ri jα1 X α ′ . Lemma 2.1 yields, Ẏ i ′ j (0, t) = -Ri jα1 X α ′ + B(εt) = -Ri j21 X 2 ′ + B(εt) = (δ i 1 δ 2 j -δ i 2 δ 1 j ) sin t + B(εt).
Integrating with respect to t, together with the initial condition (2.7), we derive

Y i ′ j (0, t) = Y i ′ j (0, 0) + ∫ t 0 Ẏ i ′ j (0, s)ds = δ i 2 δ 1 j -δ i 1 δ 2 j + (δ i 1 δ 2 j -δ i 2 δ 1 j ) ∫ t 0 sin sds + B(εt 2 ) = (δ i 2 δ 1 j -δ i 1 δ 2 j ) cos t + B(εt 2 ).
2). Similarly, we take account of Ẏ ′′ . Differentiating equation (2.6) twice with respect to the parameter θ :

Ẏ i ′′ j + ∂ 2 pq Γ i kl Ẋl X p ′ X q ′ Y k j + 2∂ p Γ i kl Ẋl ′ X p ′ Y k j + ∂ p Γ i kl Ẋl X p ′′ Y k j + 2∂ p Γ i kl Ẋl X p ′ Y k ′ j + Γ i kl ( Ẋl ′′ Y k j + 2 Ẋl ′ Y k ′ j + Ẋl Y k ′′ j ) = 0,
with the initial condition

Y i ′′ j (0, 0) = -δ i 1 δ 1 j -δ i 2 δ 2 j . (2.8)
On the axis, replacing Ẋ, Y, Γ i kl , ∂ p Γ i k1 by the corresponding values, the above equation reduces to

Ẏ i ′′ j + ∂ 2 αβ Γ i 1j X α ′ X β ′ + 2∂ α Γ i βj X α ′ Ẋβ ′ + R i jα1 X α ′′ + 2R i kα1 X α ′ Y k ′ j = 0,
As a result, the term Ẏ i ′′ j has the form

Ẏ i ′′ j = -∂ 2 αβ Γ i 1j X α ′ X β ′ -2∂ α Γ i βj X α ′ Ẋβ ′ - R i jα1 X α ′′ -2R i kα1 X α ′ Y k ′ j .
Since the result follows from Lemma 2.1 if j is equal to 1, thus it is sufficient to assume that j > 1.

From (1.17), Lemma 2.1 and (1.15), it follows that

Ẏ i ′′ j = - 2 3 (R i βαj + R i jαβ )X α ′ Ẋβ ′ -2R i kα1 X α ′ Y k ′ j + B(εt) = - 2 3 ( Ri βαj + Ri jαβ )X α ′ Ẋβ ′ -2 Ri kα1 X α ′ Y k ′ j + B(εt) = - 2 3 ( Ri 22j + Ri j22 )X 2 ′ Ẋ2 ′ -2δ 2 j Ri 121 X 2 ′ Y 1 ′ 2 + B(εt) = - 2 3 Ri 22j sin t cos t + 2δ 2 j Ri 121 sin t cos t + B(εt) = 2 3 (δ i j -δ i 2 δ 2 j ) sin t cos t + 2δ i 2 δ 2 j sin t cos t + B(εt).
Integrating with respect to t, together with the initial condition (2.8), we have

Y i ′′ j (0, t) = Y i ′′ j (0, 0) - ∫ t 0 Ẏ i ′′ j (0, s)ds = -δ i 2 δ 2 j + 2δ i 2 δ 2 j ∫ t 0 sin s cos sds + 2 3 (δ i j -δ i 2 δ 2 j ) ∫ t 0 sin s cos sds + B(εt 2 ) = -δ i 2 δ 2 j + δ i 2 δ 2 j sin 2 t + 1 3 (δ i j -δ i 2 δ 2 j ) sin 2 t + B(εt 2 ) = -δ i 2 δ 2 j cos 2 t + 1 3 (δ i j -δ i 2 δ 2 j ) sin 2 t + B(εt 2 ).
3). Differentiating equation (2.6) one with respect to t and θ respectively:

Ÿ i ′ j + ∂ 2 pq Γ i kl Ẋl Ẋp X q ′ Y k j + ∂ p Γ i kl ( Ẍl X p ′ Y k j + Ẋl Ẋp ′ Y k j + Ẋl X p ′ Ẏ k j ) + ∂ p Γ i kl ( Ẋl ′ Y k j + Ẋl Y k ′ j ) Ẋp + Γ i kl ( Ẍl ′ Y k j + Ẋl ′ Ẏ k j + Ẍl Y k ′ j + Ẋl Ẏ k ′ j ) = 0.
Evaluating at the origin, together with the following relations

X ′ (0, 0) = 0, Ẋ(0, 0) = (1, 0 • • • , 0) T , Ẋ′ (0, 0) = (0, 1, 0 • • • , 0) T , Y (0, 0) = I n , (1.15) and Γ i kl (0) = ∂ 1 Γ i kl (0) = 0, we see that Ÿ i ′ j (0, 0) = -R i j21 (0).
Differentiating equation (2.6) twice with respect to θ and once with respect to t respectively:

Ÿ i ′′ j + 2∂ 2 pq Γ i kl Ẋl Ẋp X q ′ Y k ′ j + 2∂ p Γ i kl ( Ẍl X p ′ Y k ′ j + Ẋl Ẋp ′ Y k ′ j + Ẋl X p ′ Ẏ k ′ j ) + ∂ 2 pq Γ i kl Ẋl Ẋp X q ′′ Y k j + ∂ p Γ i kl ( Ẍl X p ′′ Y k j + Ẋl Ẋp ′′ Y k j + Ẋl X p ′′ Ẏ k j ) + 2∂ 2 pq Γ i kl Ẋl ′ Ẋp X q ′ Y k j + 2∂ p Γ i kl ( Ẍl ′ X p ′ Y k j + Ẋl ′ Ẋp ′ Y k j + Ẋl ′ X p ′ Ẏ k j ) + ∂ 3 apq Γ i kl Ẋl Ẋa X p ′ X q ′ Y k j + ∂ 2 pq Γ i kl ( Ẍl X p ′ X q ′ Y k j + Ẋl Ẋp ′ X q ′ Y k j + Ẋl X p ′ Ẋq ′ Y k j + Ẋl X p ′ X q ′ Ẏ k j ) + ∂ p Γ i kl ( Ẋl ′′ Y k j + 2 Ẋl ′ Y k ′ j + Ẋl Y k ′′ j ) Ẋp + Γ i kl ( Ẍl ′′ Y k j + Ẋl ′′ Ẏ k j + 2 Ẍl ′ Y k ′ j + 2 Ẋl ′ Ẏ k ′ j + Ẍl Y k ′′ j + Ẋl Ẏ k ′′ j ) = 0,
Evaluating at the origin, combining [START_REF] Cheeger | Comparison Theorems in Riemannian Geometry[END_REF] with the relations Ẋ(0, 0

) = (1, 0 • • • , 0) T , X ′ (0, 0) = X ′′ (0) = 0, Ẋ′ (0, 0) = (0, 1, 0 • • • , 0) T , Ẋ′′ (0, 0) = (-1, 0 • • • , 0) T , Y (0, 0) = I n and Γ k ij (0) = ∂ 1 Γ k ij (0) = 0, we get Ÿ i ′′ j (0, 0) = -2R i k21 (0)Y k ′ j (0) -2∂ 2 Γ i 2j (0) = -2δ 1 j R i 221 (0) + 2δ 2 j R i 121 (0) - 2δ 1 j R i 221 (0) - 2 3 (1 -δ 1 j )R i 22j (0) = - 10 3 δ 1 j R i 221 + 2δ 2 j R i 121 (0) - 2 3 R i 22j (0) = 2δ 2 j R i 121 (0) - 2 3 (1 + 5δ 1 j )R i 22j (0).
This completes the proof of Lemma 2.2.

The behaviour of the curvature matrix

In this section, we take account of the behaviour of the curvature matrix. Combining (1.8) with γ θ (t) = e 1 (θ, t), by the anti-symmetry of the Riemann curvature tensor, we see that R ij (θ, t) = 0, if i = 1 or j = 1. Thus all order partial derivatives of R 1j and R i1 with respect to θ or t vanish identically. Without loss of generality, we assume that i, j > 1 in this section. In the following, the Riemann curvature tensor are evaluated at γ 0 (t), otherwise specifically.

The behaviour of curvature matrix is illustrated as follows.

Proposition 2.1. Let (M, g) be a closed Riemannian manifold of dimension n ≥ 2.

Assume that (M, g) satisfies the curvature assumptions (1.3) and (1.4). Fix m 0 ∈ M and ν 0 ∈ I(m 0 )\{0}. In the Fermi coordinate system, for any t ∈ [0, τ ] and i, j > 1, we have

a) Ṙij (0, t) = ∇ 1 R 1i1j , R ′ ij (0, t) = (R 1i2j + R 1j2i ) cos t + ∇ 2 R 1i1j sin t + B(εt 2 ); b) Rij (0, t) = ∇ 2 11 R 1i1j , Ṙ′ ij (0, t) = (∇ 1 R 1i2j + ∇ 1 R 1j2i + ∇ 2 R 1i1j ) cos t + B(εt), R ′′ ij (0, t) = [2R 2i2j + (δ 2i + δ 2j -2)R 1i1j ] cos 2 t+ (-∇ 1 R 1i1j + 2∇ 2 R 1i2j + 2∇ 2 R 1j2i ) sin t cos t + B(εt 2 ); c) Ṙ′′ ij (0, 0) = 2∇ 1 R 2i2j (0) + (δ 2i + δ 2j -3)∇ 1 R 1i1j (0)+ 2(∇ 2 R 1i2j (0) + ∇ 2 R 1j2i (0)), R′ ij (0, 0) = B(ε), R′′ ij (0, 0) = B(ε).
Remark 2.1. [START_REF] Alías | Maximum Principles and Geometric Applications[END_REF].We will need the initial value of Ṙ, R ′ , Ṙ′ and R ′′ . Based on a) and b), we know that

Ṙ(0, 0) = [ 0 0 0 ∇ 1 R 1i1j (0) ] , R ′ (0, 0) = [ 0 0 0 R 1i2j (0) + R 1j2i (0) ] , Ṙ′ (0, 0) = [ 0 0 0 ∇ 1 R 1i2j (0) + ∇ 1 R 1j2i (0) + ∇ 2 R 1i1j (0) ] , R ′′ (0, 0) =   0 0 0 0 0 -R 121j (0) 0 -R 1i12 (0) 2R 2i2j (0) -2R 1i1j (0)   ;
(2).From a) and b), we know that Ṙ, R ′ , R, Ṙ′ and R ′′ are all globally small enough,i.e. there exists a positive constant C under control such that

∀t ∈ [0, π), max{| Ṙ|, |R ′ |, | R|, | Ṙ′ |, |R ′′ |} ≤ Cε.
Proof. Fix i, j > 1. To differentiate R ij , we need to rewrite the expression (1.8) in the Fermi coordinate system. Recall that

e 1 (θ, t) = Ẋj (θ, t)∂ j , e i (θ, t) = Y j i (θ, t)∂ j .
From (1.8), we can write

R ij (θ, t) = ⟨R(e i (θ, t), e 1 (θ, t))e 1 (θ, t), e j (θ, t)⟩ = R dcba (X) Ẋa Y b i Ẋc Y d j = R abcd (X) Ẋa Y b i Ẋc Y d j , (2.9)
where the last equality holds with respect to the symmetry of the Riemann curvature tensor. a). Differentiating (2.9) with respect to t :

Ṙij (θ, t) = ∂ p R abcd Ẋa Y b i Ẋc Y d j Ẋp + R abcd ∂ t ( Ẋa Y b i Ẋc Y d j ) = ∂ p R abcd Ẋa Y b i Ẋc Y d j Ẋp + R abcd ( Ẍa Y b i Ẋc Y d j + (2.10) Ẋa Ẏ b i Ẋc Y d j + Ẋa Y b i Ẍc Y d j + Ẋa Y b i Ẋc Ẏ d j ).
The partial derivative in the first term can be written in terms of the covariant derivative. The definition of the first covariant derivative gives:

∂ p R abcd = ∇ p R abcd + Γ h pa R hbcd + Γ h pb R ahcd + Γ h pc R abhd + Γ h pd R abch .
Since the Christoffel symbols vanish identically on the axis, it follows that

∂ p R abcd = ∇ p R abcd . (2.11)
Evaluating on the axis, and substituting (2.11) and X(0, t) = (t, 0,

• • • , 0) T , Y (0, t) = I n into (2.10), the result is Ṙij (0, t) = ∇ 1 R 1i1j .
To finish the proof of a), it remains to estimate R ′ ij . Differentiating (2.9) with respect to θ :

R ′ ij (θ, t) = ∂ p R abcd Ẋa Y b i Ẋc Y d j X p ′ + R abcd ∂ θ ( Ẋa Y b i Ẋc Y d j ) = ∂ p R abcd Ẋa Y b i Ẋc Y d j X p ′ + R abcd ( Ẋa ′ Y b i Ẋc Y d j + Ẋa Y b ′ i Ẋc Y d j + Ẋa Y b i Ẋc ′ Y d j + Ẋa Y b i Ẋc Y d ′ j ).
Evaluating on the axis, (2.11) and the relations X(0, t) = (t, 0,

• • • , 0) T , X 1 ′ (0, t) = 0, Y (0, t) = I n infer R ′ ij (0, t) = ∇ α R 1i1j X α ′ + (R αi1j + R 1iαj ) Ẋα ′ + R 1α1j Y α ′ i + R 1i1α Y α ′ j .
Using Lemma 2.1 and Lemma 2.2, we obtain

R ′ ij (0, t) = ∇ 2 R 1i1j X 2 ′ + (R 2i1j + R 1i2j ) Ẋ2 ′ + B(εt 2 ) = ∇ 2 R 1i1j sin t + (R 1i2j + R 1j2i ) cos t + B(εt 2 ).
b) Differentiating (2.9) twice with respect to t, we have

Rij (θ, t) = ∂ 2 pq R abcd Ẋa Y b i Ẋc Y d j Ẋp Ẋq + ∂ p R abcd Ẋa Y b i Ẋc Y d j Ẍp + 2∂ p R abcd ∂ t ( Ẋa Y b i Ẋc Y d j ) Ẋp + R abcd ∂ 2 t ( Ẋa Y b i Ẋc Y d j ) = ∂ 2 pq R abcd Ẋa Y b i Ẋc Y d j Ẋp Ẋq + ∂ p R abcd Ẋa Y b i Ẋc Y d j Ẍp + (2.12) 2∂ p R abcd ( Ẍa Y b i Ẋc Y d j + Ẋa Ẏ b i Ẋc Y d j + Ẋa Y b i Ẍc Y d j + Ẋa Y b i Ẋc Ẏ d j ) Ẋp + R abcd ( Ẍa Y b i Ẋc Y d j + 2 Ẍa Ẏ b i Ẋc Y d j + 2 Ẍa Y b i Ẍc Y d j + 2 Ẍa Y b i Ẋc Ẏ d j + Ẋa Ÿ b i Ẋc Y d j + 2 Ẋa Ẏ b i Ẍc Y d j + 2 Ẋa Ẏ b i Ẋc Ẏ d j + Ẋa Y b i Ẍc Y d j + 2 Ẋa Y b i Ẍc Ẏ d j + Ẋa Y b i Ẋc Ÿ d j ).
We write the second partial derivatives in above expression in terms of the related covariant derivatives. By the definition of the second covariant derivatives, we see that

∂ 2 pq R abcd = ∇ 2 pq R abcd + ∂ p Γ h qa R hbcd + ∂ p Γ h qb R ahcd + ∂ p Γ h qc R abhd + ∂ p Γ h qd R abch + Γ h qa ∂ p R hbcd + Γ h qb ∂ p R ahcd + Γ h qc ∂ p R abhd + Γ h qd ∂ p R abch + Γ h pq ∇ h R abcd + Γ h pa ∇ q R hbcd + Γ h pb ∇ q R ahcd + Γ h pc ∇ q R abhd + Γ h pd ∇ q R abch .
Evaluating on the axis, since the the Christoffel symbols vanish identically, we have

∂ 2 pq R abcd = ∇ 2 pq R abcd + ∂ p Γ h qa R hbcd + ∂ p Γ h qb R ahcd + (2.13) ∂ p Γ h qc R abhd + ∂ p Γ h qd R abch .
Recall that ∂ 1 Γ i jk = 0 on the axis. Thus, we get

∂ 2 1q R 1i1j = ∇ 2 1q R 1i1j . (2.14) Substituting (2.14) and X(0, t) = (t, 0, • • • , 0) T , Y (0, t) = I n into (2.12), the result is Rij (0, t) = ∇ 2 11 R 1i1j .
Next, we begin to estimate Ṙ′ ij . Differentiating (2.9) with respect to θ and t respectively:

Ṙ′ ij (θ, t) = ∂ 2 pq R abcd Ẋa Y b i Ẋc Y d j Ẋq X p ′ + ∂ p R abcd Ẋa Y b i Ẋc Y d j Ẋp ′ + ∂ p R abcd ∂ t ( Ẋa Y b i Ẋc Y d j )X p ′ + ∂ p R abcd ∂ θ ( Ẋa Y b i Ẋc Y d j ) Ẋp + R abcd ∂ 2 tθ ( Ẋa Y b i Ẋc Y d j ) = ∂ 2 pq R abcd Ẋa Y b i Ẋc Y d j Ẋq X p ′ + ∂ p R abcd Ẋa Y b i Ẋc Y d j Ẋp ′ + ∂ p R abcd ( Ẍa Y b i Ẋc Y d j + Ẋa Ẏ b i Ẋc Y d j + Ẋa Y b i Ẍc Y d j + Ẋa Y b i Ẋc Ẏ d j )X p ′ + ∂ p R abcd ( Ẋa ′ Y b i Ẋc Y d j + Ẋa Y b ′ i Ẋc Y d j + Ẋa Y b i Ẋc ′ Y d j + Ẋa Y b i Ẋc Y d ′ j ) Ẋp + R abcd ( Ẍa ′ Y b i Ẋc Y d j + Ẋa ′ Ẏ b i Ẋc Y d j + Ẋa ′ Y b i Ẍc Y d j + Ẋa ′ Y b i Ẋc Ẏ d j + Ẍa Y b ′ i Ẋc Y d j + Ẋa Ẏ b ′ i Ẋc Y d j + Ẋa Y b ′ i Ẍc Y d j + Ẋa Y b ′ i Ẋc Ẏ d j + Ẍa Y b i Ẋc ′ Y d j + Ẋa Ẏ b i Ẋc ′ Y d j + Ẋa Ẏ b i Ẍc ′ Y d j + Ẋa Y b i Ẋc ′ Ẏ d j + Ẍa Y b i Ẋc Y d ′ j + Ẋa Ẏ b i Ẋc Y d ′ j + Ẋa Y b i Ẍc Y d ′ j + Ẋa Y b i Ẋc Ẏ d ′ j ).
Evaluating on the axis, the relations

X(0, t) = (t, 0, • • • , 0) T , X 1 ′ (0, t) = 0, and Y (0, t) = I n infer Ṙ′ ij (0, t) = ∂ 2 1α R 1i1j X α ′ + ∂ α R 1i1j Ẋα ′ + ∂ 1 R αi1j Ẋα ′ + ∂ 1 R 1α1j Y α ′ i + ∂ 1 R 1iαj Ẋα ′ + ∂ 1 R 1i1α Y α ′ j + R αi1j Ẍα ′ + R 1α1j Ẏ α ′ i + R 1iαj Ẍα ′ + R 1i1α Ẏ α ′ j = ∂ 2 1α R 1i1j X α ′ + (∂ α R 1i1j + ∂ 1 R αi1j + ∂ 1 R 1iαj ) Ẋα ′ + (R αi1j + R 1iαj ) Ẍα ′ + ∂ 1 R 1α1j Y α ′ i + ∂ 1 R 1i1α Y α ′ j + +R 1α1j Ẏ α ′ i + R 1i1α Ẏ α ′ j .
From (2.11), (2.14), Lemma 2.1 and Lemma 2.2, we get

Ṙ′ ij (0, t) = ∇ 2 1α R 1i1j X α ′ + (∇ α R 1i1j + ∇ 1 R αi1j + ∇ 1 R 1iαj ) Ẋα ′ + (R αi1j + R 1iαj ) Ẍα ′ + ∇ 1 R 1α1j Y α ′ i + ∇ 1 R 1i1α Y α ′ j + +R 1α1j Ẏ α ′ i + R 1i1α Ẏ α ′ j = ∇ 2 12 R 1i1j X 2 ′ + (∇ 2 R 1i1j + ∇ 1 R 2i1j + ∇ 1 R 1i2j ) Ẋ2 ′ + (R 2i1j + R 1i2j ) Ẍ2 ′ + B(εt) = ∇ 2 12 R 1i1j sin t + (∇ 2 R 1i1j + ∇ 1 R 2i1j + ∇ 1 R 1i2j ) cos t - (R 2i1j + R 1i2j ) sin t + B(εt) = (∇ 1 R 1i2j + ∇ 1 R 1j2i + ∇ 2 R 1i1j ) cos t + B(εt),
where in the last equality the fact was used that R 1i2j = B(ε).

To finish the proof of b), it remains to estimate R ′′ ij . Differentiating (2.9) twice with respect to θ, we infer

R ′′ ij (θ, t) = ∂ 2 pq R abcd Ẋa Y b i Ẋc Y d j X p ′ X q ′ + ∂ p R abcd Ẋa Y b i Ẋc Y d j X p ′′ + 2∂ p R abcd ∂ θ ( Ẋa Y b i Ẋc Y d j )X p ′ + R abcd ∂ 2 θθ ( Ẋa Y b i Ẋc Y d j ) = ∂ 2 pq R abcd Ẋa Y b i Ẋc Y d j X p ′ X q ′ + ∂ p R abcd Ẋa Y b i Ẋc Y d j X p ′′ + 2∂ p R abcd ( Ẋa ′ Y b i Ẋc Y d j + Ẋa Y b ′ i Ẋc Y d j + Ẋa Y b i Ẋc ′ Y d j + Ẋa Y b i Ẋc Y d ′ j )X p ′ + R abcd ( Ẋa ′′ Y b i Ẋc Y d j + 2 Ẋa ′ Y b ′ i Ẋc Y d j + 2 Ẋa ′ Y b i Ẋc ′ Y d j + 2 Ẋa ′ Y b i Ẋc Y d ′ j + Ẋa Y b ′′ i Ẋc Y d j + 2 Ẋa Y b ′ i Ẋc ′ Y d j + 2 Ẋa Y b ′ i Ẋc Y d ′ j + Ẋa Ẏ b i Ẋc ′′ Y d j + 2 Ẋa Y b i Ẋc ′ Y d ′ j + Ẋa Y b i Ẋc Y d ′′ j ).
Evaluated on the axis, the relations

X(0, t) = (t, 0, • • • , 0) T , X 1 ′ (0, t) = 0, Y (0, t) = I n imply R ′′ ij (0, t) = ∂ 2 αβ R 1i1j X α ′ X β ′ + ∂ p R 1i1j X p ′′ + 2∂ α R βi1j X α ′ Ẋβ ′ + 2∂ α R 1β1j X α ′ Y β ′ i + 2∂ α R 1iβj X α ′ Ẋβ ′ + 2∂ α R 1i1β X α ′ Y β ′ j + R ai1j Ẋa ′′ + 2R αb1j Ẋα ′ Y b ′ i + 2R αiβj Ẋα ′ Ẋβ ′ + 2R αi1β Ẋα ′ Y β ′ j + R 1β1j Y β ′′ i + 2R 1βαj Ẋα ′ Y β ′ i + 2R 1α1β Y α ′ i Y β ′ j + R 1iaj Ẋa ′′ + 2R 1iαb Ẋα ′ Y b ′ j + R 1i1β Y β ′′ j = ∂ 2 αβ R 1i1j X α ′ X β ′ + ∂ p R 1i1j X p ′′ + 2(∂ α R βi1j + ∂ α R 1iβj )X α ′ Ẋβ ′ + 2(∂ α R 1β1j X α ′ Y β ′ i + ∂ α R 1i1β X α ′ Y β ′ j ) + (R ai1j + R 1iaj ) Ẋa ′′ + 2R αiβj Ẋα ′ Ẋβ ′ + 2(R αb1j Ẋα ′ Y b ′ i + R 1iαb Ẋα ′ Y b ′ j ) + 2(R αi1β Ẋα ′ Y β ′ j + R 1βαj Ẋα ′ Y β ′ i ) + 2R 1α1β Y α ′ i Y β ′ j + R 1β1j Y β ′′ i + R 1i1β Y β ′′ j .
We are now in the position to calculate ∂ 2 αβ R 1i1j and ∂ p R 1i1j . For the first covariant derivative, from (2.11), we have

∂ p R 1i1j = ∇ p R 1i1j .
For the second covariant derivative, from (2.13), it follows that

∂ 2 αβ R 1i1j = ∇ 2 αβ R 1i1j + ∂ α Γ p 1β R pi1j + ∂ α Γ p iβ R 1p1j + ∂ α Γ p 1β R 1ipj + ∂ α Γ p jβ R 1i1p = ∇ 2 αβ R 1i1j + ∂ α Γ p 1β (R pi1j + R 1ipj ) + ∂ α Γ p iβ R 1p1j + ∂ α Γ p jβ R 1i1p = ∇ 2 αβ R 1i1j + R p βα1 (R 1ipj + R 1jpi ) + 1 3 (R p iαβ + R p βαi )R 1p1j + (2.15) 1 3 (R p jαβ + R p βαj )R 1i1p .
where the last equality holding due to (1.15) and (1.16).

Hence ∂ 2 αβ R 1i1j is uniformly bounded. Therefore R ′′ ij (0, t) = ∂ 2 αβ R 1i1j X α ′ X β ′ + ∇ p R 1i1j X p ′′ + 2(∇ α R βi1j + ∇ α R 1iβj )X α ′ Ẋβ ′ + 2(∇ α R 1β1j X α ′ Y β ′ i + ∇ α R 1i1β X α ′ Y β ′ j ) + (R ai1j + R 1iaj ) Ẋa ′′ + 2R αiβj Ẋα ′ Ẋβ ′ + 2(R αb1j Ẋα ′ Y b ′ i + R 1iαb Ẋα ′ Y b ′ j ) + 2(R αi1β Ẋα ′ Y β ′ j + R 1βαj Ẋα ′ Y β ′ i ) + 2R 1α1β Y α ′ i Y β ′ j + R 1β1j Y β ′′ i + R 1i1β Y β ′′ j .
From Lemma 2.1 and Lemma 2.2, we obtain

R ′′ ij (0, t) = ∂ 2 22 R 1i1j X 2 ′ X 2 ′ + ∇ 1 R 1i1j X 1 ′′ + 2(∇ 2 R 2i1j + ∇ 2 R 1i2j )X 2 ′ Ẋ2 ′ + 2R 1i1j Ẋ1 ′′ + 2R 2i2j Ẋ2 ′ Ẋ2 ′ + 2(δ 2i R 211j Ẋ2 ′ Y 1 ′ 2 + R 1i21 δ 2j Ẋ2 ′ Y 1 ′ 2 ) + R 1i1j Y i ′′ i + R 1i1j Y j ′′ j + B(εt 2 ) = ∂ 2 22 R 1i1j sin 2 t -∇ 1 R 1i1j sin t cos t + 2(∇ 2 R 2i1j + ∇ 2 R 1i2j ) sin t cos t -2R 1i1j cos(2t) + 2R 2i2j cos 2 t + 2(δ 2i + δ 2j )R 1i1j cos 2 t -(δ 2i + δ 2j )R 1i1j cos 2 t + 1 3 (δ i3 + δ j3 + • • • + δ in + δ jn )R 1i1j sin 2 t + B(εt 2 ) = ∂ 2 22 R 1i1j sin 2 t + (-∇ 1 R 1i1j + 2∇ 2 R 2i1j + 2∇ 2 R 1i2j ) sin t cos t -2R 1i1j cos(2t) + 2R 2i2j cos 2 t + (δ 2i + δ 2j )R 1i1j cos 2 t + 1 3 (δ i3 + δ j3 + • • • + δ in + δ jn )R 1i1j sin 2 t + B(εt 2 ).
For the term ∂ 2 22 R 1i1j , the formula (2.15) gives:

∂ 2 22 R 1i1j = ∇ 2 22 R 1i1j + R p 221 (R pi1j + R 1ipj ) + 1 3 R p 22i R 1p1j + 1 3 R p 22j R 1i1p = - 8 3 R 1i1j + 1 3 (δ 2i + δ 2j )R 1i1j + B(ε),
where the last equality follows from the assumption (1.4). Therefore

R ′′ ij (0, t) = - 8 3 R 1i1j sin 2 t + 1 3 (δ 2i + δ 2j )R 1i1j sin 2 t + (-∇ 1 R 1i1j + 2∇ 2 R 2i1j + 2∇ 2 R 1i2j ) sin t cos t - 2R 1i1j cos(2t) + 2R 2i2j cos 2 t + (δ 2i + δ 2j )R 1i1j cos 2 t + 1 3 (δ i3 + δ j3 + • • • + δ in + δ jn )R 1i1j sin 2 t + B(εt 2 ). Since δ i3 + • • • + δ in = 1 -δ 2i , δ j3 + • • • + δ jn = 1 -δ 2j , for i, j > 1. Then R ′′ ij (0, t) = - 8 3 R 1i1j sin 2 t + 1 3 (δ 2i + δ 2j )R 1i1j sin 2 t + (-∇ 1 R 1i1j + 2∇ 2 R 2i1j + 2∇ 2 R 1i2j ) sin t cos t - 2R 1i1j cos(2t) + 2R 2i2j cos 2 t + (δ 2i + δ 2j )R 1i1j cos 2 t + 1 3 (2 -δ 2i -δ 2j )R 1i1j sin 2 t + B(εt 2 ) = -2R 1i1j sin 2 t + (-∇ 1 R 1i1j + 2∇ 2 R 2i1j + 2∇ 2 R 1i2j ) sin t cos t -2R 1i1j cos(2t) + 2R 2i2j cos 2 t + (δ 2i + δ 2j )R 1i1j cos 2 t + B(εt 2 ) = 2(R 2i2j -R 1i1j ) cos 2 t + (-∇ 1 R 1i1j + 2∇ 2 R 1i2j + 2∇ 2 R 1j2i ) sin t cos t + (δ 2i + δ 2j )R 1i1j cos 2 t + B(εt 2 ).
c). Differentiating (2.9) twice with respect to θ and once with respect to t :

Ṙ′′ ij (θ, t) = ∂ 3 pqk R abcd Ẋa Y b i Ẋc Y d j Ẋk X p ′ X q ′ + ∂ 2 pq R abcd ∂ t ( Ẋa Y b i Ẋc Y d j )X p ′ X q ′ + ∂ 2 pq R abcd Ẋa Y b i Ẋc Y d j ( Ẋp ′ X q ′ + X p ′ Ẋq ′ ) + ∂ 2 pq R abcd Ẋa Y b i Ẋc Y d j Ẋq X p ′′ + ∂ p R abcd ∂ t ( Ẋa Y b i Ẋc Y d j )X p ′′ + ∂ p R abcd Ẋa Y b i Ẋc Y d j Ẋp ′′ + 2∂ 2 pq R abcd ∂ θ ( Ẋa Y b i Ẋc Y d j ) Ẋq X p ′ + 2∂ p R abcd ∂ 2 tθ ( Ẋa Y b i Ẋc Y d j )X p ′ + 2∂ p R abcd ∂ θ ( Ẋa Y b i Ẋc Y d j ) Ẋp ′ + ∂ p R abcd ∂ 2 θθ ( Ẋa Y b i Ẋc Y d j ) Ẋp + R abcd ∂ 3 tθθ ( Ẋa Y b i Ẋc Y d j ).
Evaluating at the origin, the initial conditions X ′ (0, 0) = X ′′ (0, 0) = 0 infer

Ṙ′′ ij (0, 0) = ∂ p R abcd Ẋa Y b i Ẋc Y d j Ẋp ′′ + 2∂ p R abcd ∂ θ ( Ẋa Y b i Ẋc Y d j ) Ẋp ′ + ∂ p R abcd ∂ 2 θθ ( Ẋa Y b i Ẋc Y d j ) Ẋp + R abcd ∂ 3 tθθ ( Ẋa Y b i Ẋc Y d j ). The term 2∂ p R abcd ∂ θ ( Ẋa Y b i Ẋc Y d j ) Ẋp ′ can be handled as follows 2∂ p R abcd ∂ θ ( Ẋa Y b i Ẋc Y d j ) Ẋp ′ = 2∂ p R abcd ( Ẋa ′ Y b i Ẋc Y d j + Ẋa Y b ′ i Ẋc Y d j + Ẋa Y b i Ẋc ′ Y d j + Ẋa Y b i Ẋc Y d ′ j ) Ẋp ′ = 2∂ α R βi1j Ẋα ′ Ẋβ ′ + 2∂ α R 1β1j Ẋα ′ Y β ′ i + 2∂ α R 1iβj Ẋα ′ Ẋβ ′ + 2∂ α R 1i1β Ẋα ′ Y β ′ j = 2(∇ 2 R 1i2j + ∇ 2 R 1j2i ).
where the last equality follows from Ẋ′ (0, 0) = (0, 1, 0,

• • • , 0) T and Y i ′ j (0, 0) = -δ i 1 δ 2 j + δ i 2 δ 1 j . The term ∂ p R abcd ∂ 2 θθ ( Ẋa Y b i Ẋc Y d j ) Ẋp . ∂ p R abcd ∂ 2 θθ ( Ẋa Y b i Ẋc Y d j ) Ẋp = ∂ p R abcd ( Ẋa ′′ Y b i Ẋc Y d j + 2 Ẋa ′ Y b ′ i Ẋc Y d j + 2 Ẋa ′ Y b i Ẋc ′ Y d j + 2 Ẋa ′ Y b i Ẋc Y d ′ j + Ẋa Y b ′′ i Ẋc Y d j + 2 Ẋa Y b ′ i Ẋc ′ Y d j + 2 Ẋa Y b ′ i Ẋc Y d ′ j + Ẋa Ẏ b i Ẋc ′′ Y d j + 2 Ẋa Y b i Ẋc ′ Y d ′ j + Ẋa Y b i Ẋc Y d ′′ j ) Ẋp = ∂ 1 R ai1j Ẋa ′′ + 2∂ 1 R αb1j Ẋα ′ Y b ′ i + 2∂ 1 R αiβj Ẋα ′ Ẋβ ′ + 2∂ 1 R αi1β Ẋα ′ Y β ′ j + ∂ 1 R 1β1j Y β ′′ i + 2∂ 1 R 1βαj Ẋα ′ Y β ′ i + 2∂ 1 R 1α1β Y α ′ i Y β ′ j + ∂ 1 R 1iaj Ẋa ′′ + 2∂ 1 R 1iαb Ẋα ′ Y b ′ j + ∂ 1 R 1i1β Y β ′′ j = -2∇ 1 R 1i1j + (δ 2i + δ 2j )∇ 1 R 1i1j + 2∇ 1 R 2i2j ,
where the last equality follows from the following relations Ẋ′ (0, 0) = (0, 1, 0,

• • • , 0) T , Ẍ′ (0, 0) = (0, -1, 0, • • • , 0) T , Ẋ′′ (0, 0) = (-1, 0, • • • , 0) T and Y i ′ j (0, 0) = -δ i 1 δ 2 j + δ i 2 δ 1 j . The term ∂ 3 tθθ ( Ẋa Y b i Ẋc Y d j ). ∂ 3 tθθ ( Ẋa Y b i Ẋc Y d j ) = Ẍa ′′ Y b i Ẋc Y d j + Ẋa ′′ Ẏ b i Ẋc Y d j + Ẋa ′′ Y b i Ẍc Y d j + Ẋa ′′ Y b i Ẋc Ẏ d j + 2 Ẍa ′ Y b ′ i Ẋc Y d j + 2 Ẋa ′ Ẏ b ′ i Ẋc Y d j + 2 Ẋa ′ Y b ′ i Ẍc Y d j + 2 Ẋa ′ Y b ′ i Ẋc Ẏ d j + 2 Ẍa ′ Y b i Ẋc ′ Y d j + 2 Ẋa ′ Ẏ b i Ẋc ′ Y d j + 2 Ẋa ′ Y b i Ẍc ′ Y d j + 2 Ẋa ′ Y b i Ẋc ′ Ẏ d j + 2 Ẍa ′ Y b i Ẋc Y d ′ j + 2 Ẋa ′ Ẏ b i Ẋc Y d ′ j + 2 Ẋa ′ Y b i Ẍc Y d ′ j + 2 Ẋa ′ Y b i Ẋc Ẏ d ′ j + Ẍa Y b ′′ i Ẋc Y d j + Ẋa Ẏ b ′′ i Ẋc Y d j + Ẋa Y b ′′ i Ẍc Y d j + Ẋa Y b ′′ i Ẋc Ẏ d j + 2 Ẍa Y b ′ i Ẋc ′ Y d j + 2 Ẋa Ẏ b ′ i Ẋc ′ Y d j + 2 Ẋa Y b ′ i Ẍc ′ Y d j + 2 Ẋa Y b ′ i Ẋc ′ Ẏ d j + 2 Ẍa Y b ′ i Ẋc Y d ′ j + 2 Ẋa Ẏ b ′ i Ẋc Y d ′ j + 2 Ẋa Y b ′ i Ẍc Y d ′ j + 2 Ẋa Y b ′ i Ẋc Ẏ d ′ j + Ẍa Y b i Ẋc ′′ Y d j + Ẋa Ẏ b i Ẋc ′′ Y d j + Ẋa Y b i Ẍc ′′ Y d j + Ẋa Y b i Ẋc ′′ Ẏ d j + 2 Ẍa Y b i Ẋc ′ Y d ′ j + 2 Ẋa Ẏ b i Ẋc ′ Y d ′ j + 2 Ẋa Y b i Ẍc ′ Y d ′ j + 2 Ẋa Y b i Ẋc ′ Ẏ d ′ j + Ẍa Y b i Ẋc Y d ′′ j + Ẋa Ẏ b i Ẋc Y d ′′ j + Ẋa Y b i Ẍc Y d ′′ j + Ẋa Y b i Ẋc Ẏ d ′′ j = 0,
where the last equality holds because of Ẍ(0, 0) = Ẍ′ (0, 0) = Ẍ′′ (0, 0) = 0 and Ẏ (0, 0

) = Ẏ ′ (0, 0) = Ẏ ′′ (0, 0) = 0. Therefore Ṙ′′ ij (0, 0) = 2∇ 1 R 2i2j + (δ 2i + δ 2j -3)∇ 1 R 1i1j + 2(∇ 2 R 1i2j + ∇ 2 R 1j2i ).
By a lengthy computation, one can prove that R′ ij (0, 0) = B(ε) and R′′ ij (0, 0) = B(ε). This ends the proof of Proposition 2.1.

Chapter 3 A reinforced MTW condition

This chapter contains the various subtle estimates of the M T W tensor. The M T W tensor plays an important role in the regularity of the optimal transport map. Thus it is useful to understand the behaviour of the M T W tensor. Firstly, we exploit the formula in [START_REF] Figalli | Nearly round spheres look convex[END_REF] to calculate the M T W tensor. Secondly, we recast the approximation of the M T W tensor by the M T W tensor on the round sphere originated from Theorem 2 in [START_REF] Delanoë | Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds[END_REF]. In the end, we prove that the M T W tensor on the nearly spherical manifold satisfies a reinforced M T W condition.

The calculation of MTW tensor

In this section, we calculate the M T W tensor. As a starting point, we give some notations. Let (M, g) be an n-dimensional Riemannian manifold. Henceforth, we fix m ∈ M, ν ∈ I(m)\{0} and (ξ, η) ∈ T m M ×T m M. Since the M T W tensor is homogeneous with degree 2 in both ξ and η, it suffices to assume that |ξ| m = |η| m = 1.

Taking the orthonormal basis

{E 1 , E 2 , • • • , E n } of the tangent space T m M so that ν = |ν| m E 1 , ξ = ξ 1 E 1 + ξ 2 E 2 + ξ 3 E 3 , η = η 1 E 1 + η 2 E 2
and identify the tangent vectors at m with their coordinates in this basis. Then the metric g m is given by the canonical scalar product of R n . It will be implicitly understood throughout the calculations that the inner product and the Riemann curvature tensor are evaluated at the point m.

Recalling the definition of M T W tensor (0.2), it follows that

C(m, ν)(ξ, η) = - 3 2 d 2 ds 2 s=0 ⟨∇ 2 m c(., exp m (ν + sη))(ξ), ξ⟩ m = - 3 2 d 2 ds 2 s=0 ⟨S(m, ν + sη, 1)(ξ), ξ⟩ m ,
where the second equality follows from Proposition 1.1(b).

For any s ∈ R small enough, we can write

ν + sη = t(s)(cos θE 1 + sin θE 2 ),
where

t(s) = |ν + sη| m , θ(s) = tan -1 ( sη 2 τ + sη 1
) .

From the Remark 1.1(1), it follows that

C(m, ν)(ξ, η) = - 3 2 d 2 ds 2 | s=0 ⟨S(m, cos θ(s)E 1 + sin θ(s)E 2 , t(s))(ξ), ξ⟩ m .
To proceed, we give some more notations. Let γ θ (t) be the geodesic with initial point m and initial velocity cos θE 1 + sin θE 2 . For |θ| sufficiently small, let the orthonormal frame {e 1 (θ, t), e 2 (θ, t), • • • , e n (θ, t)} be the parallel transport along the geodesic 25 γ θ (t) with the initial conditions e 1 (θ, 0) = cos θE 1 + sin θE 2 , e 2 (θ, 0) =sin θE 1 + cos θE 2 , e i (θ, 0) = E i for i ≥ 3. Let J 0 (θ, t) and J 1 (θ, t) be the solutions of the Jacobi equation (1.7) and R(θ, t) be defined by (1.8) along the geodesic γ θ (t). Recall that the matrix of the linear operator S(m, e 1 (θ, 0), t) in the orthonormal basis {e 1 (θ, 0), e 2 (θ, 0),

• • • , e n (θ, 0)} is tJ 0 (θ, t) -1 J 1 (θ, t).
Then the matrix of the linear operator S(m, e 1 (θ, 0), t) in the orthonormal basis

{E 1 , E 2 , • • • , E n } is Q(θ) T S(θ, t)Q(θ), i.e. ⟨S(m, e 1 (θ, 0), t)(ξ), ξ⟩ = ⟨S(θ, t)Q(θ)ξ, Q(θ)ξ⟩. (3.1)
where

Q(θ) =   cos θ sin θ -sin θ cos θ I n-2   .
In the sequel, the dot will stand for the derivative with respect to t and the prime for the derivative with respect to θ.

Differentiating (3.1) once and twice with respect to s, respectively:

d ds ⟨S(m, e 1 (θ, 0), t)(ξ), ξ⟩ = [⟨S ′ Qξ, Qξ⟩ + 2⟨SQξ, Q ′ ξ⟩] dθ ds + ⟨ ṠQξ, Qξ⟩ dt ds , d 2 ds 2 ⟨S(m, e 1 (θ, 0), t)(ξ), ξ⟩ = (3.2) [⟨S ′′ Qξ, Qξ⟩ + 4⟨S ′ Qξ, Q ′ ξ⟩ + 2⟨SQ ′ ξ, Q ′ ξ⟩ + 2⟨SQξ, Q ′′ ξ⟩]( dθ ds ) 2 + [2⟨ Ṡ′ Qξ, Qξ⟩ + 4⟨ ṠQξ, Q ′ ξ⟩] dt ds dθ ds + ⟨ SQξ, Qξ⟩( dt ds ) 2 + [⟨S ′ Qξ, Qξ⟩ + 2⟨SQξ, Q ′ ξ⟩] d 2 θ ds 2 + ⟨ ṠQξ, Qξ⟩ d 2 t ds 2 .
By a straitforward computation, we have

dt ds = s(η 2 1 + η 2 2 ) + τ η 1 |ν + sη| , d 2 t ds 2 = τ 2 η 2 2 |ν + sη| 3 , dθ ds = τ η 2 s 2 (η 2 1 + η 2 2 ) + 2sτ η 1 + τ 2 , d 2 θ ds 2 = - 2τ η 2 [s(η 2 1 + η 2 2 ) + τ η 1 ] [s 2 (η 2 1 + η 2 2 ) + 2sτ η 1 + τ 2 ] 2 ,
where τ = |ν| m .

Evaluating at s = 0, we have

t = τ, dt ds = η 1 , d 2 t ds 2 = η 2 2 τ , θ = 0, dθ ds = η 2 τ , d 2 θ ds 2 = - 2η 1 η 2 τ 2 . (3.3)
We write

P ξ = (ξ 1 , ξ 2 , 0, • • • , 0) T , P ⊥ ξ = (ξ 2 , -ξ 1 , 0, • • • , 0) T .
Then at s = 0, we have

Qξ = ξ, Q ′ ξ = P ⊥ ξ, Q ′′ ξ = -P ξ. (3.4)
At s = 0, plugging (3.3),(3.4) into (3.2), we obtain

d 2 ds 2 | s=0 ⟨S(m, e 1 (θ, 0), t)(ξ), ξ⟩ = [⟨S ′′ ξ, ξ⟩ + 4⟨S ′ ξ, P ⊥ ξ⟩ + 2⟨SP ⊥ ξ, P ⊥ ξ⟩ -2⟨Sξ, P ξ⟩] η 2 2 τ 2 + [2⟨ Ṡ′ ξ, ξ⟩ + 4⟨ Ṡξ, P ⊥ ξ⟩] η 1 η 2 τ + ⟨ Sξ, ξ⟩η 2 1 + [⟨S ′ ξ, ξ⟩ + 2⟨Sξ, P ⊥ ξ⟩](- 2η 1 η 2 τ 2 ) + ⟨ Ṡξ, ξ⟩ η 2 2 τ = ⟨ Sξ, ξ⟩η 2 1 + [ 2 τ ⟨ Ṡ′ ξ, ξ⟩ + 4 τ ⟨ Ṡξ, P ⊥ ξ⟩ - 2 τ 2 ⟨S ′ ξ, ξ⟩ - 4 τ 2 ⟨Sξ, P ⊥ ξ⟩]η 1 η 2 + [ 1 τ 2 ⟨S ′′ ξ, ξ⟩ + 1 τ ⟨ Ṡξ, ξ⟩ + 4 τ 2 ⟨S ′ ξ, P ⊥ ξ⟩ + 2 τ 2 ⟨SP ⊥ ξ, P ⊥ ξ⟩ - 2 τ 2 ⟨Sξ, P ξ⟩]η 2 2 .
Finally, we get

C(m, ν)(ξ, η) = - 3 2 ⟨ Sξ, ξ⟩η 2 1 + 3[- 1 τ ⟨ Ṡ′ ξ, ξ⟩ - 2 τ ⟨ Ṡξ, P ⊥ ξ⟩+ (3.5) 1 τ 2 ⟨S ′ ξ, ξ⟩ + 2 τ 2 ⟨Sξ, P ⊥ ξ⟩]η 1 η 2 + 3 2 [- 1 τ 2 ⟨S ′′ ξ, ξ⟩- 1 τ ⟨ Ṡξ, ξ⟩ - 4 τ 2 ⟨S ′ ξ, P ⊥ ξ⟩ - 2 τ 2 ⟨SP ⊥ ξ, P ⊥ ξ⟩ + 2 τ 2 ⟨Sξ, P ξ⟩]η 2 2 = a 11 (m, ν, ξ)η 2 1 + a 12 (m, ν, ξ)η 1 η 2 + a 22 (m, ν, ξ)η 2 2 .
We note that the term ξ 2 1 does not appear in the coefficients a 11 and a 12 . The details will be discussed in section 3.3.

The approximation of MTW tensor

In this section, we recast Theorem 2 in [START_REF] Delanoë | Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds[END_REF] as follows.

Theorem 3.1. Let (M, g) be a closed n-dimensional Riemannian manifold satisfying (1.3) and (1.4)

for some positive constant ε. Given m ∈ M and ν ∈ I(m). Assume ε is small enough such that |ν| sin |ν| ε ≤ 1 4 √ n -1 .
Then there exists a positive constant C ≥ 1 (independent of (m, ν, ε)) such that, for any unit tangent vectors ξ, η ∈ T m M, the following inequality holds:

|C(m, ν)(ξ, η) -C(m, ν)(ξ, η)| ≤ Cε( |ν| sin |ν| ) 4 (|ξ ⊥ | 2 + |η ⊥ | 2 ).
We give some comments about Theorem 3.1. Theorem 3.1 provides a qualified control of M T W tensor by the M T W tensor on the sphere. The control depends on ε and |ν| which will be important for the blow up rates when |ν| is close to π. It will be used to show that, under the hypothesis of (1.3) and (1.4), the M T W tensor satisfies a reinforced M T W condition in section 3.3. Remark 3.1. Theorem 3.1 is obvious when ν is equal to 0, because of ( 4) and [START_REF] Caffarelli | A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity[END_REF]. Therefore, we will consider only the case ν ∈ I(m)\{0}.

Proof of Theorem 3.1. Recall that we can calculate the M T W tensor by formula (3.5). Replacing the term C by C, we get the corresponding formula for MTW tensor on the sphere C. The difference between C and C can be estimated as follows:

C(m, ν)(ξ, η) -C(m, ν)(ξ, η) (3.6) = - 3 2 ⟨( S -S)ξ, ξ⟩η 2 1 + 3[- 1 τ ⟨ Ṡ′ ξ, ξ⟩ + 1 τ 2 ⟨S ′ ξ, ξ⟩ - 2 τ ⟨( Ṡ - Ṡ)ξ, P ⊥ ξ⟩ + 2 τ 2 ⟨(S -S)ξ, P ⊥ ξ⟩]η 1 η 2 + 3 2 [- 1 τ 2 ⟨S ′′ ξ, ξ⟩ - 1 τ ⟨( Ṡ -Ṡ)ξ, ξ⟩ - 4 τ 2 ⟨S ′ P ⊥ ξ, ξ⟩ - 2 τ 2 ⟨(S -S)P ⊥ ξ, P ⊥ ξ⟩ + 2 τ 2 ⟨(S -S)ξ, P ξ⟩]η 2 2 .
Observe we calculate the above expression on the right hand side at the point (m, E 1 , τ ) where τ = |ν|.

The proof is divided into four steps. From now on, we will compute at the point (m, E 1 , t) where t ∈ (0, τ ].

Step 1.In first step, we will prove that there exist a positive constant C such that,∀t ∈ (0, τ ],

|S -S| ≤ Cε t 6 sin 4 t , (3.7) | Ṡ -Ṡ| ≤ Cε t 5 sin 4 t , ( 3.8)
| S -S| ≤ Cε t 4 sin 4 t . ( 3.9)
By integrating with respect to t, it suffices to show the last inequality (3.9). We start by evaluating the difference J -1 0 -J-1 0 . For this purpose, we write

J 0 = J0 [I n -J-1 0 ( J0 -J 0 )].
Then we infer the formal expansion

J -1 0 = ∞ ∑ k=0 [ J-1 0 ( J0 -J 0 )] k J-1 0 .
Recalling that J0 is diagonal and first row and first column of the matrix J0 -J 0 vanish, we have

J -1 0 -J-1 0 = t sin t ∞ ∑ k=1 ( t sin t ) k ( J0 -J 0 ) k . From Remark 1.2, we derive that |J 0 -J0 | ≤ 2ε √ n -1.
Together with the assumption

τ sin τ ε ≤ 1 4 √
n-1 and the real function t sin t is increase in the interval (0, π), the latter expansion is uniformly convergent in any compact subset in (0, π). Moreover, we have

|J -1 0 -J-1 0 | ≤ 4 √ n -1( t sin t ) 2 ε, (3.10)
provided ε is sufficiently small. The triangle inequality provides the upper bound

|(J -1 0 ) ⊥ | ≤ 2 √ n -1 t sin t . (3.11)
We are in position to estimate the term S -S. The second derivative of S with respect to t takes the form

S = 2J -1 0 J1 -2tJ -1 0 J0 J -1 0 J1 + 2tJ -1 0 J0 J -1 0 J0 J -1 0 J 1 -2J -1 0 J0 J -1 0 J 1 . (3.12)
Replacing S by S, one can get the corresponding formula. After using the finite differences trick in a systematic way, we have

S -S = 2(J -1 0 -J-1 0 ) J1 + 2 J-1 0 ( J1 -J1 ) -2t(J -1 0 -J-1 0 ) J0 J -1 0 J1 - 2t J-1 0 ( J0 -J0 )J -1 0 J1 -2t J-1 0 J0 (J -1 0 -J-1 0 ) J1 - 2t J-1 0 J0 J-1 0 ( J1 -J1 ) + 2t(J -1 0 -J-1 0 ) J0 J -1 0 J0 J -1 0 J 1 + 2t J-1 0 ( J0 -J0 )J -1 0 J0 J -1 0 J 1 + 2t J-1 0 J0 (J -1 0 -J-1 0 ) J0 J -1 0 J 1 + 2t J-1 0 J0 J-1 0 ( J0 -J0 )J -1 0 J 1 + 2t J-1 0 J0 J-1 0 J0 (J -1 0 -J-1 0 )J 1 + 2t J-1 0 J0 J-1 0 J0 J-1 0 (J 1 -J1 ) -2t(J -1 0 -J-1 0 ) J0 J -1 0 J1 - 2t J-1 0 ( J0 -J0 )J -1 0 J1 -2t J-1 0 J0 (J -1 0 -J-1 0 ) J1 - 2t J-1 0 J0 J-1 0 ( J1 -˙J1 ) + t(J -1 0 -J-1 0 ) J1 + t J-1 1 ( J1 -J1 ).
From (3.10), (3.11), Lemma 1.1 and Lemma 1.2, we infer that there exsits a positive constant C such that

| S -S| ≤ Cε t 4 sin 4 t , ∀t ∈ (0, τ ]. (3.13)
Step 2.In this step, we will show that there exists a positive constant C > 0 such that

|S ′ | ≤ Cε t 4 sin 2 t , (3.14) | Ṡ′ | ≤ Cε t 4 sin 3 t , ∀t ∈ (0, τ ]. (3.15)
We first deal with J ′ 0 and J ′ 1 . By differentiating the equation (1.7) with respect to the variable θ, evaluating at the point (0, t), one can derive that J ′ 0 and J ′ 1 satisfy the following equations

{ J′ a + RJ ′ a = -R ′ J a , J ′
a (0) = 0 = J′ a (0), a = 0, 1. By the representation formula (Proposition 1.1 (c)), we obtain

J ′ a = -J 0 ∫ t 0 J * 1 R ′ J a ds + J 1 ∫ t 0 J * 0 R ′ J a ds, a = 0, 1. (3.16)
By virtue of a) in Proposition 2.1 and Lemma 1.1, we derive that there exists a positive constant C such that (3.17) and Lemma 1.1, we infer that there exists a positive constant C such that

|J ′ a | ≤ Cεt, a = 0, 1. (3.17) Note that S ′ = -tJ -1 0 J ′ 0 J -1 0 J 1 + tJ -1 0 J ′ 1 . From (3.11),
|S ′ | ≤ Cε t 4 sin 2 t .
We are in position to estimate (3.15). We first handle J′ 0 and J′ 1 . By differentiating the equation (1.7) with respect to θ and t once respectively, evaluated at the point (0, t), one can derive that J′ 0 and J′ 1 satisfy the following equations

     J′ a + R J′ a = -Ṙ′ J a -R ′ Ja -ṘJ ′ a , a = 0, 1. J′ 0 (0) = 0 = J′ 0 (0), J′ 1 (0) = 0, J′ 1 (0) = -R ′ (0).
From the representation formula (Proposition 1.1 (c)), we have

J′ 0 = -J 0 ∫ t 0 J * 1 ( Ṙ′ J 0 + R ′ J0 + ṘJ ′ 0 )ds + J 1 ∫ t 0 J * 0 ( Ṙ′ J 0 + R ′ J0 + ṘJ ′ 0 )ds, (3.18)
J′ 1 = -J 0 R ′ (0) -J 0 ∫ t 0 J * 1 ( Ṙ′ J 1 + R ′ J1 + ṘJ ′ 1 )ds + J 1 ∫ t 0 J * 0 ( Ṙ′ J 1 + R ′ J1 + ṘJ ′ 1 )ds. (3.19)
By virtue of a), b) in Proposition 2.1 and Lemma 1.1, we derive that there exists a positive constant C such that

| J′ 0 | ≤ Cεt, | J′ 1 | ≤ Cε. Note that Ṡ′ = -J -1 0 J ′ 0 J -1 0 J 1 + tJ -1 0 J0 J -1 0 J ′ 0 J -1 0 J 1 - (3.20) tJ -1 0 J′ 0 J -1 0 J 1 + tJ -1 0 J ′ 0 J -1 0 J0 J -1 0 J 1 - tJ -1 0 J ′ 0 J -1 0 J1 + J -1 0 J ′ 1 -tJ -1 0 J0 J -1 0 J ′ 1 + tJ -1 0 J′ 1 .
Together with (3.17), (3.11) and Lemma 1.1, we infer that there exists a positive constant C such that

| Ṡ′ | ≤ Cε t 4 sin 3 t .
Step 3.In this step, we will verify that there exists a positive constant C such that

∀t ∈ (0, τ ], |S ′′ | ≤ Cε t 5 sin 3 t . ( 3.21)
We first deal with J ′′ 0 and J ′′ 1 . Differentiating the equation (1.7) with respect to θ twice, evaluating at the point (0, t), one can derive that J ′′ 0 and J ′′ 1 satisfy the following equations

{ J′′ a + RJ ′′ a = -R ′′ J a -2R ′ J ′ a , a = 0, 1, J ′′ a (0) = 0 = J′′ a (0)
. Making use of the representation formula (Proposition 1.1 (c)), we see that

J ′′ a = -J 0 ∫ t 0 J * 1 (R ′′ J a + 2R ′ J ′ a )ds + J 1 ∫ t 0 (R ′′ J a + 2R ′ J ′ a )ds, a = 0, 1. (3.22)
By virtue of a), b) in Proposition 2.1 and Lemma 1.1, we derive that there exists a positive constant C such that

|J ′′ 0 | ≤ Cεt, |J ′′ 1 | ≤ Cεt.
Note that

S ′′ = 2tJ -1 0 J ′ 0 J -1 0 J ′ 0 J -1 0 J 1 -tJ -1 0 J ′′ 0 J -1 0 J 1 - (3.23) 2tJ -1 0 J ′ 0 J -1 0 J ′ 1 + tJ -1 0 J ′′ 1 .
Together with (3.17), (3.11) and Lemma 1.1, we infer that there exists a positive constant C such that

|S ′′ | ≤ Cε t 5 sin 3 t , ∀t ∈ (0, τ ].
Step 4. Notice that the first row and the first column of the matrices S -S, Ṡ′ , S ′ , Ṡ -Ṡ, S -S, S ′′ all vanish. Using (3.6), (3.7), (3.8), (3.9), (3.14), (3.15), (3.21) and the fact

that |ξ 2 2 η 1 η 2 |, |ξ 2 ξ 3 η 1 η 2 |, |ξ 2 3 η 1 η 2 |, |ξ 1 ξ 2 η 1 η 2 |, |ξ 1 ξ 3 η 1 η 2 | are all controlled by |ξ ⊥ | 2 +|η ⊥ | 2
, we get the desired result. This completes the proof of Theorem 3.1.

A reinforced MTW condition

This section is devoted to showing that the M T W tensor on nearly spherical manifold has a reinforced lower bound. The result is formulated as follows.

Theorem 3.2. Let (M, g) be a closed n-dimensional Riemannian manifold satisfying the curvature assumption (1.3). Then there exist some positive constants ε 0 and κ 0 such that if

∥Riem - 1 2 g g∥ C 2 (M,g) < ε 0 .
Then for any m ∈ M, ν ∈ I(m) and any tangent vectors ξ, η in T m M,

C(m, ν)(ξ, η) ≥ κ 0 (|ξ ∧ η| 2 m + |ξ| 2 m |η ∧ ν| 2 m + |ξ ∧ ν| 2 m |η| 2 m ), ( 3
= |ξ| 2 m |η| 2 m -g m (ξ, η) 2 , |η ∧ν| 2 m = |η| 2 m |ν| 2 m -g m (η, ν) 2 , |ξ ∧ ν| 2 m = |ξ| 2 m |ν| 2 m -g m (ξ, ν) 2 .
Notice that Theorem 3.2 was established for n = 2 in [START_REF] Delanoë | Locally nearly spherical surfaces are almost-positively curved[END_REF]. Moreover, the round sphere S n satisfies the curvature assumptions in Theorem 3.2, the associated result was established in [START_REF] Delanoë | Positively curved Riemannian locally symmetric spaces are positively squared distance curved[END_REF]. The similar result was obtained under the C 4 perturbation of the round spheres in [START_REF] Figalli | Nearly round spheres look convex[END_REF]. It is easy to see that Theorem 3.2 implies that the closed Riemannian manifold under the above hypotheses satisfies the A3S condition.

As a direct consequence of Theorem 3.2, the A3W condition holds on Riemannian product of nearly spherical manifolds. Corallary 3.1. Let M 1 and M 2 be two closed Riemannian manifolds of dimension n 1 ≥ 2 and n 2 ≥ 2 respectively. If M 1 and M 2 both satisfy the curvature assumptions as in Theorem 3.2, then the A3W condition holds on the Riemannian product manifold M 1 × M 2 . Moreover, the associated MTW tensor is non-negative.

Proof of Theorem 3.2. Remark the tangent vector ν takes value in I(m). To prove Theorem 3.2, we discuss three cases for MTW tensor: when the tangent vector ν is close to the origin, close to the focalization and rest cases. In the following, the length of the tangent vector ν is denoted by τ.

Step 1.We consider the behaviour of M T W tensor when ν is away from the zero and from the focalization.

Assume that 0

< δ 1 ≤ τ ≤ δ 2 < t F (m, ν) ≤ π and 0 < ε < sin δ2 4δ2 √ n-1 . Since the function t sin t is non-decreasing in the interval [0, π), thus the condition of Theorem 3.1 is satisfied. Then C(m, ν)(ξ, η) ≥ C(m, ν)(ξ, η) -C( δ 2 sin δ 2 ) 4 ε(ξ 2 2 + ξ 2 3 + η 2 2 ).
As it was known that, on the round sphere, there exists a positive constant [START_REF] Delanoë | Positively curved Riemannian locally symmetric spaces are positively squared distance curved[END_REF] such that

C(m, ν)(ξ, η) ≥ κ0 [ (|ξ| 2 |η| 2 -⟨ξ, η⟩ 2 ) + τ 2 (ξ 2 2 + ξ 2 3 )|η| 2 + τ 2 |ξ| 2 η 2 2 ] = κ0 [ (|ξ| 2 |η| 2 -⟨ξ, η⟩ 2 ) + τ 2 (ξ 2 2 + ξ 2 3 + η 2 2 )
] .

As a direct consequence, we have

C(m, ν)(ξ, η) ≥ κ0 (|ξ| 2 |η| 2 -⟨ξ, η⟩ 2 ) + (κ 0 τ 2 - Cδ 4 2 sin 4 δ 2 ε)(ξ 2 2 + ξ 2 3 + η 2 2 ) ≥ κ0 (|ξ| 2 |η| 2 -⟨ξ, η⟩ 2 ) + (κ 0 - Cδ 4 2 δ 2 1 sin 4 δ 2 ε)τ 2 (ξ 2 2 + ξ 2 3 + η 2 2 ) ≥ κ0 2 [|ξ| 2 |η| 2 -⟨ξ, η⟩ 2 + τ 2 (ξ 2 2 + ξ 2 3 + η 2 2 )].
where the last inequality follows from the assumption ε < κ0δ 2 1 sin 4 δ2 2Cδ 4

2

.

Then for δ 1 ≤ τ ≤ δ 2 , we infer the existence of constant

ε 1 = κ0δ 2 1 sin 4 δ2 2Cδ 4 2 and κ 1 = κ0 2 such that C(m, ν)(ξ, η) ≥ κ 1 [ (|ξ| 2 |η| 2 -⟨ξ, η⟩ 2 ) + τ 2 (ξ 2 2 + ξ 2 3 )|η| 2 + τ 2 |ξ| 2 η 2 2
] .

Step 2. We investigate the behaviour of M T W tensor near the focalization. Under the curvature assumption (1.3) and (1.4)(ε is small enough), one can derive by the method in [START_REF] Figalli | Nearly round spheres look convex[END_REF] near the focalization that there exist positive numbers ε 2 , κ 2 , δ2 ( 3π 4 ≤ δ2 < π) such that the M T W tensor has the following estimate:

ε < ε 2 , τ ≥ δ2 , C(m, ν)(ξ, η) ≥ κ 2 (|S ⊥ ξ| 2 |η| 2 + ξ 2 1 η 2 2 ), (3.25)
where S ⊥ denotes the orthogonal projection of S on the orthogonal subspace ν ⊥ .

Recalling the curvature assumption (1.3), then the Hessian comparison theorem [START_REF] Chavel | Eigenvalues in Riemannian Geometry[END_REF] infers

-S ⊥ ≥ - τ cos τ sin τ I n-1 .
Hence the term

|S ⊥ ξ| 2 controls ξ 2 2 + ξ 2 3 if τ ≥ 3π 4 , i.e. |S ⊥ ξ| 2 ≥ 2(ξ 2 2 + ξ 2 3
). In addition, by Cauchy-Schwarz inequality, the term [START_REF] Delanoë | Gradient rearrangement for diffeomorphisms of a compact manifold[END_REF], up to the constant κ 2 , we see that

|ξ 1 ξ 2 η 1 η 2 | is bounded by ξ 2 1 η 2 2 + 1 4 ξ 2 2 η 2 1 . From (3.
C(m, ν)(ξ, η) ≥ κ 2 [ 1 2 ξ 2 1 η 2 2 + 1 4 ξ 2 2 η 2 1 - 1 2 ξ 1 ξ 2 η 1 η 2 + (ξ 2 2 + ξ 2 3 )|η| 2 ] = κ 2 [ 1 4 (|ξ| 2 |η| 2 -⟨ξ, η⟩ 2 ) + 1 4 ξ 2 1 η 2 2 + ξ 2 2 |η| 2 + 3 4 ξ 2 3 |η| 2 ] ≥ κ 2 [ (|ξ| 2 |η| 2 -⟨ξ, η⟩ 2 ) + (ξ 2 2 + ξ 2 3 )|η| 2 + |ξ| 2 η 2 2 ] ≥ κ 2 [ (|ξ| 2 |η| 2 -⟨ξ, η⟩ 2 ) + τ 2 (ξ 2 2 + ξ 2 3 )|η| 2 + τ 2 |ξ| 2 η 2 2 ] ,
where the last inequality holding due to τ < π.

We derive the existence of constant ε 2 , κ 2 and δ2 such that for 0

< ε < ε 2 , δ2 ≤ τ < t F (m, ν) C(m, ν)(ξ, η) ≥ κ 2 [ (|ξ| 2 |η| 2 -⟨ξ, η⟩ 2 ) + τ 2 (ξ 2 2 + ξ 2 3 )|η| 2 + τ 2 |ξ| 2 η 2 2
] .

Step 3.In the last step, we examine the asymptotic behaviour of M T W tensor when ν is near the origin. From (3.5), we know that it suffices to study the expansion of the coefficients a 11 , a 12 and a 22 . By definition, the coefficients a 11 , a 12 and a 22 contain S, Ṡ, S ′ , Ṡ′ , S, S ′′ . As a result, we only need to deal with the expansion of S, Ṡ, S ′ , Ṡ′ , S, S ′′ . It is clear that S 1 j (θ, t) = S j 1 (θ, t) = δ 1 j . Without loss of generality, we address the expansion of S i j , Ṡi j , S i ′ j , Ṡi ′ j , Si j , S i ′′ j with i, j ≥ 2. In the following, given a real function f, we write f = O(εt p + t q )(p, q > 0) if there exists two positive constant µ 0 , C which are both independent of ε such that ∀t ∈ [0, µ 0 ), |f (t)| ≤ C(εt p + t q ). The notation f = O(εt p )(p > 0) or f = O(t q )(q > 0) can be defined in the similar way.

We first take account of the asymptotic behavior of the coefficient a 11 which involves S.

Lemma 3.1. Under the curvature assumptions (1.3) and (1.4), on the axis, we have

S(0, t) = I n - t 2 3 R(0, 0) - t 3 12 Ṙ(0, 0) - t 4 45 A + O(εt 4 + t 6 ), (3.26) Ṡ(0, t) = - 2 3 tR(0, 0) - t 2 4 Ṙ(0, 0) - 4 45 t 3 A + O(εt 3 + t 5 ), (3.27) S(0, t) = - 2 3 R(0, 0) - t 2 Ṙ(0, 0) - 4 15 t 2 A + O(εt 2 + t 4 ). (3.28)
where A =

[ 0 0 0 I n-1
] .

Remark 3.2. [START_REF] Alías | Maximum Principles and Geometric Applications[END_REF].The formula (3.26) can also recover the expression of the M T W tensor in the special case ν = 0. Using a Riemannian normal coordinate system at m, it follows from the definition of the M T W tensor (0.2) that

C(m, 0)(ξ, η) = - 3 2
d 2 ds 2 | s=0 ⟨S(m, η, s)(ξ), ξ⟩ = ⟨R(0, 0)ξ, ξ⟩ = R m (ξ, η, ξ, η).
Before showing Lemma 3.1, we give some facts about the coefficient a 11 . Since the first row and first column of S vanish, the coefficient a 11 is independent of ξ 2 1 . As a consequence of (3.28), one has the expansion of a 11

a 11 (m, ν, ξ) = ⟨R(0, 0)ξ, ξ⟩ + 3 4 τ ⟨ Ṙ(0, 0)ξ, ξ⟩ + 2 5 τ 2 ⟨Aξ, ξ⟩ + O(ετ 2 + τ 4 )(ξ 2 2 + ξ 2 3 ) = R 1212 ξ 2 2 + 2R 1213 ξ 2 ξ 3 + R 1313 ξ 2 3 + 3 4 τ (∇ 1 R 1212 ξ 2 2 + 2∇ 1 R 1213 ξ 2 ξ 3 + ∇ 1 R 1313 ξ 2 3 ) + 2 5 τ 2 (ξ 2 2 + ξ 2 3 ) + O(ετ 2 + τ 4 )(ξ 2 2 + ξ 2 3 ) = R(ξ, E 1 , ξ, E 1 ) + 3 4 τ (∇ 1 R 1212 ξ 2 2 + 2∇ 1 R 1213 ξ 2 ξ 3 + (3.29) ∇ 1 R 1313 ξ 2 3 ) + 2 5 τ 2 (ξ 2 2 + ξ 2 3 ) + O(ετ 2 + τ 4 )(ξ 2 2 + ξ 2 3 ).
Proof of Lemma 3.1. First, we examine the expansion of S. Let J 0 and J 1 be the solutions of the following second order differential equations:

Ja (t) + R(0, 0)J a (t) = 0, a = 0, 1.
with the initial conditions J 0 (0) = 0, J0 (0

) = I n , J 1 (0) = I n , J1 (0) = 0.
Obviously, the matrices J 0 and J 1 satisfy the equations Ja (t) + R(0, 0)J a (t) = (R(0, 0) -R(0, t))J a (t), a = 0, 1.

Using the representation formula (Proposition 1.1 (c)), we derive

J 0 = J 0 + J 0 ∫ t 0 J * 1 [R(0, 0) -R(0, s)]J 0 ds -J 1 ∫ t 0 J * 0 [R(0, 0) -R(0, s)]J 0 ds, J 1 = J 1 + J 0 ∫ t 0 J * 1 [R(0, 0) -R(0, s)]J 1 ds -J 1 ∫ t 0 J * 0 [R(0, 0) -R(0, s)]J 1 ds.
Notice that the matrices J 0 and J 1 have the following expansions

J 0 (t) = tI n - t 3 6 R(0, 0) + t 5 120 R 2 (0, 0) + O(t 7 ), J 1 (t) = I n - t 2 2 R(0, 0) + t 4 24 R 2 (0, 0) + O(t 6 ).
In addition, from the Taylor formula and b) in Proposition 2.1, we see that

R(0, t) = R(0, 0) + t Ṙ(0, 0) + ∫ t 0 (t -s) R(0, s)ds, = R(0, 0) + t Ṙ(0, 0) + ∫ t 0 (t -s)∇ 2 11 R(0, s)ds, = R(0, 0) + t Ṙ(0, 0) + O(εt 2 ).
Thus

J 0 (0, t) = J 0 (t) - t 4 12 Ṙ(0, 0) + O(εt 5 ) = tI n - t 3 6 R(0, 0) - t 4 12 Ṙ(0, 0) + t 5 120 R 2 (0, 0) + O(εt 5 + t 7 ) = tI n - t 3 6 R(0, 0) - t 4 12 Ṙ(0, 0) + t 5 120 A + O(εt 5 + t 7 ), (3.30
)

J 1 (0, t) = J 1 (t) - t 3 6 Ṙ(0, 0) + O(εt 4 ) = tI n - t 2 2 R(0, 0) - t 3 6 Ṙ(0, 0) + t 4 24 R 2 (0, 0) + O(εt 4 + t 6 ) = I n - t 2 2 R(0, 0) - t 3 6 Ṙ(0, 0) + t 4 24 A + O(εt 4 + t 6 ). (3.31)
Then

S(0, t) = tJ 0 (0, t) -1 J 1 (0, t) = t[tI n - t 3 6 R(0, 0) - t 4 12 Ṙ(0, 0)) + t 5 120 A + O(εt 5 + t 7 )] -1 [I n - t 2 2 R(0, 0) - t 3 6 Ṙ(0, 0) + t 4 24 A + O(εt 4 + t 6 )] = [I n + t 2 6 R(0, 0) + t 3 12 Ṙ(0, 0) + 7 360 t 4 A + O(εt 4 + t 6 )][I n - t 2 2 R(0, 0) - t 3 6 Ṙ(0, 0) + t 4 24 A + O(εt 4 + t 6 )] = I n - t 2 3 R(0, 0) - t 3 12 Ṙ(0, 0) - t 4 45 A + O(εt 4 + t 6 ).
We are in position to deal with the expansion of Ṡ. It is easy to see that Ṡ =

(I n -tJ -1 0 J0 )J -1 0 J 1 + tJ -1 0 J1 which involves J -1 0 J0 and J -1 0 J1 . We shall consider the expansions of J -1 0 J0 and J -1 0 J1 . Differentiating (1.7) with respect to t,      Ja + R Ja = -ṘJ a , a = 0, 1 J0 (0) = I n , J0 (0) = 0, J1 (0) = 0, J1 (0) = -R(0, 0).
Making use of the representation formula (Proposition 1.1 (c)) again,

J0 = J 1 -J 0 ∫ t 0 J * 1 ṘJ 0 ds + J 1 ∫ t 0 J * 0 ṘJ 0 ds, J1 = -J 0 R(0, 0) -J 0 ∫ t 0 J * 1 ṘJ 1 ds + J 1 ∫ t 0 J * 0 ṘJ 1 ds. Then tJ -1 0 J0 = tJ -1 0 J 1 -t ∫ t 0 J * 1 ṘJ 0 ds + tJ -1 0 J 1 ∫ t 0 J * 0 ṘJ 0 ds = S(0, t) -t ∫ t 0 J * 1 ṘJ 0 ds + S(0, t) ∫ t 0 J * 0 ṘJ 0 ds, (3.32) J -1 0 J1 = -R(0, 0) - ∫ t 0 J * 1 ṘJ 1 ds + J -1 0 J 1 ∫ t 0 J * 0 ṘJ 1 ds = -R(0, 0) - ∫ t 0 J * 1 ṘJ 1 ds + J -1 0 J 1 ∫ t 0 J * 0 ṘJ 1 ds. (3.33) By b) in Proposition 2.1, it follows that Ṙ(0, t) = Ṙ(0, 0) + ∫ t 0 R(0, s)ds = Ṙ(0, 0) + ∫ t 0 ∇ 2 11 R(0, s)ds = Ṙ(0, 0) + O(εt).
Then

tJ -1 0 J0 (0, t) = S(0, t) - t 3 2 Ṙ(0, 0) + S(0, t) t 3 3 Ṙ(0, 0) + O(εt 4 ) = I n - t 2 3 R(0, 0) - t 3 4 Ṙ(0, 0) - t 4 45 A + O(εt 4 + t 6 ), J -1 0 J1 (0, t) = -R(0, 0) - t 2 Ṙ(0, 0) + O(εt 2 + t 4 ).
Therefore, we obtain

Ṡ = (I n -tJ -1 0 J0 )J -1 0 J 1 + tJ -1 0 J1 = [ t 2 3 R(0, 0) + t 3 4 Ṙ(0, 0) + t 4 45 A + O(εt 4 + t 6 )][tI n - t 3 6 R(0, 0) - t 4 12 Ṙ(0, 0) + t 5 120 A + O(εt 5 + t 7 )] -1 [I n - t 2 2 R(0, 0) - t 3 6 Ṙ(0, 0) + t 4 24 A + O(εt 4 + t 6 )] + t[-R(0, 0) - t 2 Ṙ(0, 0) + O(εt 2 + t 4 )] = [ t 3 R(0, 0) + t 2 4 Ṙ(0, 0) + t 3 45 A + O(εt 3 + t 5 )][I n + t 2 6 R(0, 0) + t 3 12 Ṙ(0, 0) + 7t 4 360 A + O(εt 4 + t 6 )][I n - t 2 2 R(0, 0) - t 3 6 Ṙ(0, 0) + t 4 24 A + O(εt 4 + t 6 )] -tR(0, 0) - t 2 2 Ṙ(0, 0) + O(εt 3 + t 5 ) = [ t 3 R(0, 0) + t 2 4 Ṙ(0, 0) + 7t 3 90 A + O(εt 3 + t 5 )][I n - t 2 2 R(0, 0) - t 3 6 Ṙ(0, 0) + t 4 24 A + O(εt 4 + t 6 )] -tR(0, 0) - t 2 2 Ṙ(0, 0) + O(εt 3 + t 5 ) = t 3 R(0, 0) + t 2 4 Ṙ(0, 0) - 4t 3 45 A -tR(0, 0) - t 2 2 Ṙ(0, 0) + O(εt 3 + t 5 ) = - 2 3 tR(0, 0) - t 2 4 Ṙ(0, 0) - 4 45 t 3 A + O(εt 3 + t 5 ).
Similarly, from (3.12), we know that

S = 2(I n -tJ -1 0 J0 )J -1 0 J1 + 2(tJ -1 0 J0 ) 1 t 2 (tJ -1 0 J0 -I n )S = 2[ t 2 3 R(0, 0) + t 3 4 Ṙ(0, 0) + t 4 45 A + O(εt 4 + t 6 )][-R(0, 0)- t 2 Ṙ(0, 0) + O(εt 2 + t 4 )] + 2[I n - t 2 3 R(0, 0) - t 3 4 Ṙ(0, 0) - t 4 45 A + O(εt 4 + t 6 )][- 1 3 R(0, 0) - t 4 Ṙ(0, 0) - t 2 45 A + O(εt 2 + t 4 )][I n - t 2 3 R(0, 0) - t 3 12 Ṙ(0, 0) - t 4 45 A + O(εt 4 + t 6 )] = - 2 3 t 2 R 2 (0) + O(εt 2 + t 4 ) + 2[- 1 3 R(0, 0) - t 4 Ṙ(0, 0) + 4t 2 45 A + O(εt 2 + t 4 )][I n - t 2 3 R(0, 0) - t 3 12 Ṙ(0, 0) - t 4 45 A + O(εt 4 + t 6 )] = - 2 3 t 2 R 2 (0) - 2 3 R(0, 0) - t 2 Ṙ(0, 0) + 2 5 t 2 A + O(εt 2 + t 4 ) = - 2 3 R(0, 0) - t 2 Ṙ(0, 0) - 4 15 t 2 A + O(εt 2 + t 4 ).
This ends the proof of Lemma 3.1.

We are in position to consider the expansion of the coefficient a 12 . It is easy to see that the coefficient a 12 involves the terms S ′ and Ṡ′ . We shall examine the expansions of S ′ and Ṡ′ . Lemma 3.2. Under the curvature assumptions (1.3) and (1.4), on the axis, we have

S ′ (0, t) = - t 2 3 R ′ (0, 0) - t 3 12 Ṙ′ (0, 0) + O(εt 4 + t 6 ), (3.34)
Ṡ′ (0, t) = - 2 3 tR ′ (0, 0) - t 2 4 Ṙ′ (0, 0) + O(εt 3 + t 5 ). (3.35)
As a consequence of Lemma 3.2, we can derive the expansion of the coefficient a 12 . Noting that ⟨ξ, P ξ⟩ = 0, thus the coefficient a 12 takes the form

a 12 (m, ν, ξ) = - 3 τ ⟨ Ṡ′ ξ, ξ⟩ - 6 τ ⟨ Ṡξ, P ⊥ ξ⟩ + 3 τ 2 ⟨S ′ ξ, ξ⟩ + 6 τ 2 ⟨(S -I n )ξ, P ⊥ ξ⟩.
Since the first row and first column of Ṡ′ , Ṡ, S ′ and S -I n vanish, the coefficient a 12 is also independent of ξ

a 12 (m, ν, ξ) = 2⟨R ′ (0, 0)ξ, ξ⟩ + 3 4 τ ⟨ Ṙ′ (0, 0)ξ, ξ⟩ + 4⟨R(0, 0)ξ, P ⊥ ξ⟩ + 3 2 τ ⟨ Ṙ(0, 0)ξ, P ⊥ ξ⟩ + 8 15 τ 2 ⟨Aξ, P ⊥ ξ⟩ -⟨R ′ (0, 0)ξ, ξ⟩ - τ 4 ⟨ Ṙ′ (0, 0)ξ, ξ⟩ - 2⟨R(0, 0)ξ, P ⊥ ξ⟩ - τ 2 ⟨ Ṙ(0, 0)ξ, P ⊥ ξ⟩ - 2 15 τ 2 ⟨Aξ, P ⊥ ξ⟩ + O(εt 2 + t 4 )(ξ 2 2 + ξ 2 3 + ξ 1 ξ 2 + ξ 1 ξ 3 ) = ⟨R ′ (0, 0)ξ, ξ⟩ + 2⟨R(0, 0)ξ, P ⊥ ξ⟩ + +τ ( 1 2 ⟨ Ṙ′ (0, 0)ξ, ξ⟩ + ⟨ Ṙ(0, 0)ξ, P ⊥ ξ⟩) + 2 5 τ 2 ⟨Aξ, P ⊥ ξ⟩ + O(εt 2 + t 4 )(ξ 2 2 + ξ 2 3 + ξ 1 ξ 2 + ξ 1 ξ 3 ).
Using (1) in Remark 2.1, we get that

a 12 (m, ν, ξ) = 2R 1223 ξ 2 ξ 3 + 2R 1323 ξ 2 3 + 2(-R 1212 ξ 1 ξ 2 -R 1213 ξ 1 ξ 3 ) + τ [ 1 2 ∇ 2 R 1212 ξ 2 2 + (∇ 1 R 1223 + ∇ 2 R 1213 )ξ 2 ξ 3 + (∇ 1 R 1323 + 1 2 ∇ 2 R 1313 )ξ 2 3 -∇ 1 R 1212 ξ 1 ξ 2 -∇ 1 R 1213 ξ 1 ξ 3 ] - 2 5 τ 2 ξ 1 ξ 2 + O(ετ 2 + τ 4 )(ξ 2 2 + ξ 2 3 + ξ 1 ξ 2 + ξ 1 ξ 3 ) = 2R(ξ, E 1 , ξ, E 2 ) + τ [ 1 2 ∇ 2 R 1212 ξ 2 2 + (∇ 1 R 1223 + (3.36) ∇ 2 R 1213 )ξ 2 ξ 3 + (∇ 1 R 1323 + 1 2 ∇ 2 R 1313 )ξ 2 3 - ∇ 1 R 1212 ξ 1 ξ 2 -∇ 1 R 1213 ξ 1 ξ 3 ] - 2 5 τ 2 ξ 1 ξ 2 + O(ετ 2 + τ 4 )(ξ 2 2 + ξ 2 3 + ξ 1 ξ 2 + ξ 1 ξ 3 ).
Proof of Lemma 3.2. First, we handle the expansion of S ′ . One can prove that S ′ = -tJ -1 0 J ′ 0 J -1 0 J 1 + tJ -1 0 J ′ 1 which involves the terms J -1 0 J ′ 0 and J -1 0 J ′ 1 , thus we shall investigate the expansion of these two terms.

Recall that we can use the representations (3.16) which contain the term R ′ for the matrices J ′ 0 and J ′ 1 . We shall address the expansion of R ′ . Notice that the second order derivative R′ (0, 0) of R ′ can be computed in the following way R′ (0, 0) = 2 lim

t→0 + R ′ (0, t) -R ′ (0, 0) -t Ṙ′ (0, 0) t 2 .
By the definition of the limit, for any µ > 0, there exists a small positive constant δ > 0 which depends on µ such that

∀t ∈ [0, δ), |R ′ (0, t) -R ′ (0, 0) -t Ṙ′ (0, 0)| ≤ ( 1 2 | R′ (0, 0)| + µ)t 2 .
From c) in Proposition 2.1, we derive that there exists a positive constant C > 0 such that

∀t ∈ [0, δ), |R ′ (0, t) -R ′ (0, 0) -t Ṙ′ (0, 0)| ≤ (Cε + µ)t 2 . Assume that ε < µ, ∀t ∈ [0, δ), |R ′ (0, t) -R ′ (0, 0) -t Ṙ′ (0, 0)| ≤ (C + 1)µt 2 .
That is

R ′ (0, t) = R ′ (0, 0) + t Ṙ′ (0, 0) + O(µt 2 ).
Without the confusion, we write

R ′ (0, t) = R ′ (0, 0) + t Ṙ′ (0, 0) + O(εt 2 ). (3.37)
It follows that

J -1 0 J ′ 0 (0, t) = - ∫ t 0 J * 1 R ′ J 0 ds + J -1 0 J 1 ∫ t 0 J * 0 R ′ J 0 ds = - t 2 6 R ′ (0, 0) - t 3 12 Ṙ′ (0, 0) + O(εt 4 + t 6 ), (3.38)
J -1 0 J ′ 1 (0, t) = - ∫ t 0 J * 1 R ′ J 1 ds + J -1 0 J 1 ∫ t 0 J * 0 R ′ J 1 ds = - t 2 R ′ (0, 0) - t 2 6 Ṙ′ (0, 0) + O(εt 3 + t 5 ). (3.39)
As a consequence,

S ′ = -tJ -1 0 J ′ 0 J -1 0 J 1 + tJ -1 0 J ′ 1 = -J -1 0 J ′ 0 S + tJ -1 0 J ′ 1 = -[- t 2 6 R ′ (0, 0) - t 3 12 Ṙ′ (0, 0) + O(εt 4 + t 6 )][I n - t 2 3 R(0, 0) - t 3 12 Ṙ(0, 0) - t 4 45 A + O(εt 4 + t 6 )] - t 2 2 R ′ (0, 0) - t 3 6 Ṙ′ (0, 0) + O(εt 4 + t 6 ) = t 2 6 R ′ (0, 0) + t 3 12 Ṙ′ (0, 0) - t 2 2 R ′ (0) - t 3 6 Ṙ′ (0, 0) + O(εt 4 + t 6 ) = - t 2 3 R ′ (0, 0) - t 3 12 Ṙ′ (0, 0) + O(εt 4 + t 6 ).
Secondly, we deal with the expansion of Ṡ′ . Since Ṡ′ involves J -1 0 J′ 0 and J -1 0 J′ 1 , thus we shall investigate the expansions of these two terms.

We use again the representations (3.18) and (3.19) for J′ 0 and J′ 1 . Therefore, from c) in Proposition 2.1, the first order derivative formula R′ (0, 0) = lim t→0 + Ṙ′ (0,t)-Ṙ′ (0,0) t of Ṙ′ and by the definition of the limit, we get Ṙ′ (0, t) = Ṙ′ (0, 0) + O(εt).

(3.40)

Combining (3.30) (3.38) and (3.39), we know that

J ′ 0 (0, t) = - t 3 6 R ′ (0, 0) - t 4 12 Ṙ′ (0, 0) + O(εt 5 + t 7 ), J ′ 1 (0, t) = - t 2 2 R ′ (0, 0) - t 3 6 Ṙ′ (0, 0) + O(εt 4 + t 6 ).
Together with (3.30)(3.31) (3.37)(3.40) and Ṙ(0, t) = Ṙ(0, 0) + O(εt), it follows that

J -1 0 J′ 0 = - ∫ t 0 J * 1 ( Ṙ′ J 0 + R ′ J0 + ṘJ ′ 0 )ds + J -1 0 J 1 ∫ t 0 J * 0 ( Ṙ′ J 0 + R ′ J0 + ṘJ ′ 0 )ds = - t 2 R ′ (0, 0) - t 2 3 Ṙ′ (0, 0) + O(εt 3 + t 5 ), (3.41)
J -1 0 J′ 1 = -R ′ (0, 0) - ∫ t 0 J * 1 ( Ṙ′ J 1 + R ′ J1 + ṘJ ′ 1 )ds + J -1 0 J 1 ∫ t 0 J * 0 ( Ṙ′ J 1 + R ′ J1 + ṘJ ′ 1 )ds = -R ′ (0, 0) - t 2 Ṙ′ (0, 0) + O(εt 2 + t 4 ). (3.42)
From(3.20), we obtain

Ṡ′ = -J -1 0 J ′ 0 J -1 0 J 1 + tJ -1 0 J0 J -1 0 J ′ 0 J -1 0 J 1 - tJ -1 0 J′ 0 J -1 0 J 1 + tJ -1 0 J ′ 0 J -1 0 J0 J -1 0 J 1 - tJ -1 0 J ′ 0 J -1 0 J1 + J -1 0 J ′ 1 -tJ -1 0 J0 J -1 0 J ′ 1 + tJ -1 0 J′ 1 = (tJ -1 0 J0 -I n )J -1 0 J ′ 0 J -1 0 J 1 -J -1 0 J′ 0 S + J -1 0 J ′ 0 J -1 0 J0 S -tJ -1 0 J ′ 0 J -1 0 J1 + (I n -tJ -1 0 J0 )J -1 0 J ′ 1 + tJ -1 0 J′ 1 .
Plugging

Ṡ′ (0, t) = [- t 2 3 R(0, 0) - t 3 4 Ṙ(0, 0) - t 4 45 A + O(εt 4 + t 6 )][- t 6 R ′ (0, 0)- t 2 12 Ṙ′ (0, 0) + O(εt 3 + t 5 )][I n - t 2 3 R(0, 0) - t 3 12 Ṙ(0, 0) - t 4 45 A + O(εt 4 + t 6 )] -[- t 2 R ′ (0, 0) - t 2 3 Ṙ′ (0, 0) + O(εt 3 + t 5 )][I n - t 2 3 R(0, 0) - t 3 12 Ṙ(0, 0) - t 4 45 A + O(εt 4 + t 6 )] + [- t 6 R ′ (0, 0) - t 2 12 Ṙ′ (0, 0) + O(εt 3 + t 5 )][I n - t 2 3 R(0, 0) - t 3 4 Ṙ(0, 0) - t 4 45 A + O(εt 4 + t 6 )][I n - t 2 3 R(0, 0) - t 3 12 Ṙ(0, 0) - t 4 45 A + O(εt 4 + t 6 )] - t[- t 2 6 R ′ (0, 0) - t 3 12 Ṙ′ (0, 0) + O(εt 4 + t 6 )][-R(0, 0) - t 2 Ṙ(0, 0) + O(εt 2 + t 4 )] + [ t 2 3 R(0, 0) + t 3 4 Ṙ(0, 0) + t 4 45 A + O(εt 4 + t 6 )] [- t 2 R ′ (0, 0) - t 2 6 Ṙ′ (0, 0) + O(εt 3 + t 5 )] -tR ′ (0, 0) - t 2 2 Ṙ′ (0, 0) + O(εt 3 + t 5 ) = t 2 R ′ (0, 0) + t 2 3 Ṙ′ (0, 0) + [- t 6 R ′ (0, 0) - t 2 12 Ṙ′ (0, 0) + O(εt 3 + t 5 )] [I n - t 2 3 R(0, 0) - t 3 12 Ṙ(0, 0) - t 4 45 A + O(εt 4 + t 6 )] -tR ′ (0, 0) - t 2 2 Ṙ′ (0, 0) + O(εt 3 + t 5 ) = - t 6 R ′ (0, 0) - t 2 12 Ṙ′ (0, 0) - t 2 R ′ (0, 0) - t 2 6 Ṙ′ (0, 0) + O(εt 3 + t 5 ) = - 2 3 tR ′ (0, 0) - t 2 4 Ṙ′ (0, 0) + O(εt 3 + t 5 ).
This ends the proof of Lemma 3.2. We now consider the expansion of the coefficient a 22 . The coefficient a 22 involves the term S ′′ . We shall examine the expansion of S ′′ .

Lemma 3.3. Under the curvature assumption (1.3) and (1.4), on the axis, we have

S ′′ (0, t) = - t 2 3 R ′′ (0, 0) - t 3 12 Ṙ′′ (0, 0) + O(εt 4 + t 6 ). (3.43)
As a consequence of Lemma 3.3, we can derive the expansion of the coefficient a 22 . Noting that ⟨P ⊥ ξ, P ⊥ ξ⟩ = ⟨ξ, P ξ⟩, the coefficient a 22 takes the form

a 22 (m, ν, ξ) = - 3 2τ 2 ⟨S ′′ ξ, ξ⟩ - 3 2τ ⟨ Ṡξ, ξ⟩ - 6 τ 2 ⟨S ′ ξ, P ⊥ ξ⟩ - 3 τ 2 ⟨(S -I n )P ⊥ ξ, P ⊥ ξ⟩ + 3 τ 2 ⟨(S -I n )ξ, P ξ⟩]η 2 2 .
Note that, through the first row and first column of S ′′ , Ṡ, Ṡ′ , S -I n and S -I n vanish, the coefficient a 22 depends on ξ

a 22 (m, ν, ξ) = -R 1213 ξ 2 ξ 3 + (-R 1313 + R 2323 )ξ 2 3 + R 1212 ξ 2 2 + 2R 1213 ξ 2 ξ 3 + R 1313 ξ 2 3 -2R 1223 ξ 1 ξ 3 + R 1212 ξ 2 1 -(R 1212 ξ 2 2 + R 1213 ξ 2 ξ 3 ) + τ { 1 8 [-∇ 1 R 1212 ξ 2 2 + 4(-∇ 1 R 1213 + ∇ 2 R 1223 )ξ 2 ξ 3 + (2∇ 1 R 2323 -3∇ 1 R 1313 + 4∇ 2 R 1323 )ξ 2 3 ] + 3 8 (∇ 1 R 1212 ξ 2 2 + 2∇ 1 R 1213 ξ 2 ξ 3 + ∇ 1 R 1313 ξ 2 3 ) - 1 2 [∇ 2 R 1212 ξ 1 ξ 2 + (∇ 1 R 1223 + ∇ 2 R 1213 )ξ 1 ξ 3 ] + 1 4 ∇ 1 R 1212 ξ 2 1 - 1 4 (∇ 1 R 1212 ξ 2 2 + ∇ 1 R 1213 ξ 2 ξ 3 )} τ 2 15 (2ξ 2 2 + 2ξ 2 3 + ξ 2 1 -ξ 2 2 ) + O(ετ 2 + τ 4 ).
After combing the similar terms, we have

a 22 (m, ν, ξ) = R 1212 ξ 2 1 -2R 1223 ξ 1 ξ 3 + R 2323 ξ 2 3 + τ [ 1 4 ∇ 1 R 1212 ξ 2 1 - 1 2 ∇ 2 R 1212 ξ 1 ξ 2 - 1 2 (∇ 1 R 1223 + ∇ 2 R 1213 )ξ 1 ξ 3 + 1 2 ∇ 2 R 1223 ξ 2 ξ 3 + ( 1 4 ∇ 1 R 2323 + 1 2 ∇ 2 R 1323 )ξ 2 3 ] + τ 2 15 (ξ 2 1 + ξ 2 2 + 2ξ 2 3 ) + O(ετ 2 + τ 4 ) = R(ξ, E 2 , ξ, E 2 ) + τ [ 1 4 ∇ 1 R 1212 ξ 2 1 - 1 2 ∇ 2 R 1212 ξ 1 ξ 2 - (3.44) 1 2 (∇ 1 R 1223 + ∇ 2 R 1213 )ξ 1 ξ 3 + 1 2 ∇ 2 R 1223 ξ 2 ξ 3 + ( 1 4 ∇ 1 R 2323 + 1 2 ∇ 2 R 1323 )ξ 2 3 ] + τ 2 15 (ξ 2 1 + ξ 2 2 + 2ξ 2 3 ) + O(ετ 2 + τ 4 ).
Proof of Lemma 3.3. In view of (3.23), we know that S ′′ involves the term J -1 0 J ′′ 0 and J -1 0 J ′′ 1 . We shall investigate the expansions of these two terms. From the representations (3.22) for J ′′ 0 and J ′′ 1 , together with the Remark 2.1 and the formula R′′ (0, 0) = 2 lim t→0 + R ′′ (0,t)-R ′′ (0,0)-t Ṙ′′ (0,0) t 2 , we get R ′′ (0, t) = R ′′ (0, 0) + t Ṙ′′ (0, 0) + O(εt 2 ).

Then

J -1 0 J ′′ 0 (0, t) = - ∫ t 0 J * 1 (R ′′ J 0 + 2R ′ J ′ 0 )ds + J -1 0 J 1 ∫ t 0 (R ′′ J 0 + 2R ′ J ′ 0 )ds = - t 2 6 R ′′ (0, 0) - t 3 12 Ṙ′′ (0, 0) + O(εt 4 + t 6 ), (3.45
)

J -1 0 J ′′ 1 (0, t) = - ∫ t 0 J * 1 (R ′′ J 1 + 2R ′ J ′ 1 )ds + J -1 0 J 1 ∫ t 0 (R ′′ J 1 + 2R ′ J ′ 1 )ds = - t 2 R ′′ (0, 0) - t 2 6 Ṙ′′ (0, 0) + O(εt 3 + t 5 ). (3.46)
From (3.23), we have

S ′′ = 2J -1 0 J ′ 0 J -1 0 J ′ 0 S -J -1 0 J ′′ 0 S - 2tJ -1 0 J ′ 0 J -1 0 J ′ 1 + tJ -1 0 J ′′ 1 .
Plugging

S ′′ = 2[- t 2 6 R ′ (0, 0) - t 3 12 Ṙ′ (0, 0) + O(εt 4 + t 6 )][- t 2 6 R ′ (0, 0) - t 3 12 Ṙ′ (0, 0) + O(εt 4 + t 6 )][I n - t 2 3 R(0, 0) - t 3 12 Ṙ(0, 0) - t 4 45 A + O(εt 4 + t 6 )] -[- t 2 6 R ′′ (0, 0) - t 3 12 Ṙ′′ (0, 0) + O(εt 4 + t 6 )][I n - t 2 3 R(0, 0) - t 3 12 Ṙ(0, 0) - t 4 45 A + O(εt 4 + t 6 )] -2t[- t 2 6 R ′ (0, 0) - t 3 12 Ṙ′ (0, 0) + O(εt 4 + t 6 )][- t 2 R ′ (0, 0) - t 2 6 Ṙ′ (0, 0) + O(εt 3 + t 5 )] - t 2 2 R ′′ (0, 0) - t 3 6 Ṙ′′ (0, 0) + O(εt 4 + t 6 ) = t 2 6 R ′′ (0, 0) + t 3 12 Ṙ′′ (0, 0) - t 2 2 R ′′ (0, 0) - t 3 6 Ṙ′′ (0, 0) + O(εt 4 + t 6 ) = - t 2 3 R ′′ (0, 0) - t 3 12 Ṙ′′ (0, 0) + O(εt 4 + t 6 )
.

This ends the proof of Lemma 3.3.

From now on, we come back to the proof of Theorem 3.2. Under the above preparations, we can obtain the expansion of the M T W tensor near the origin. From (3.29)(3.36)(3.44), we have

C(m, ν)(ξ, η) = [R(ξ, E 1 , ξ, E 1 ) + 3 4 τ (∇ 1 R 1212 ξ 2 2 + 2∇ 1 R 1213 ξ 2 ξ 3 + ∇ 1 R 1313 ξ 2 3 ) + 2 5 τ 2 (ξ 2 2 + ξ 2 3 ) + O(ετ 2 + τ 4 )(ξ 2 2 + ξ 2 3 )]η 2 1 + {2R(ξ, E 1 , ξ, E 2 ) + τ [ 1 2 ∇ 2 R 1212 ξ 2 2 + (∇ 1 R 1223 + ∇ 2 R 1213 )ξ 2 ξ 3 + (∇ 1 R 1323 + 1 2 ∇ 2 R 1313 )ξ 2 3 - ∇ 1 R 1212 ξ 1 ξ 2 -∇ 1 R 1213 ξ 1 ξ 3 ] - 2 5 τ 2 ξ 1 ξ 2 + O(ετ 2 + τ 4 )(ξ 2 2 + ξ 2 3 + ξ 1 ξ 2 + ξ 1 ξ 3 )}η 1 η 2 + {R(ξ, E 2 , ξ, E 2 ) + τ [ 1 4 ∇ 1 R 1212 ξ 2 1 - 1 2 ∇ 2 R 1212 ξ 1 ξ 2 - 1 2 (∇ 1 R 1223 + ∇ 2 R 1213 )ξ 1 ξ 3 + 1 2 ∇ 2 R 1223 ξ 2 ξ 3 + ( 1 4 ∇ 1 R 2323 + 1 2 ∇ 2 R 1323 )ξ 2 3 ] + τ 2 15 (ξ 2 1 + ξ 2 2 + 2ξ 2 3 ) + O(ετ 2 + τ 4 )}η 2 2 = R(ξ, E 1 , ξ, E 1 )η 2 1 + 2R(ξ, E 1 , ξ, E 2 )η 1 η 2 + R(ξ, E 2 , ξ, E 2 )η 2 2 + 3 4 τ (∇ 1 R 1212 ξ 2 2 + 2∇ 1 R 1213 ξ 2 ξ 3 + ∇ 1 R 1313 ξ 2 3 )η 2 1 + τ [ 1 2 ∇ 2 R 1212 ξ 2 2 + (∇ 1 R 1223 + ∇ 2 R 1213 )ξ 2 ξ 3 + (∇ 1 R 1323 + 1 2 ∇ 2 R 1313 )ξ 2 3 -∇ 1 R 1212 ξ 1 ξ 2 -∇ 1 R 1213 ξ 1 ξ 3 ]η 1 η 2 + τ [ 1 4 ∇ 1 R 1212 ξ 2 1 - 1 2 ∇ 2 R 1212 ξ 1 ξ 2 - 1 2 (∇ 1 R 1223 + ∇ 2 R 1213 )ξ 1 ξ 3 + 1 2 ∇ 2 R 1223 ξ 2 ξ 3 + ( 1 4 ∇ 1 R 2323 + 1 2 ∇ 2 R 1323 )ξ 2 3 ]η 2 2 + τ 2 ( 1 15 ξ 2 1 η 2 2 - 2 5 ξ 1 ξ 2 η 1 η 2 + 2 5 ξ 2 2 η 2 1 + 1 15 ξ 2 2 η 2 2 + 2 5 ξ 2 3 η 2 1 + 2 15 ξ 2 3 η 2 2 ) + O(ετ 2 + τ 4 )(ξ 1 ξ 2 η 1 η 2 + ξ 1 ξ 3 η 1 η 2 + ξ 2 2 η 2 1 + ξ 2 2 η 1 η 2 + ξ 2 3 η 2 1 + ξ 2 3 η 1 η 2 + η 2 2 ) = I + II + III + IV.
We will estimate each term from I to IV. The combination of the zero order term and the second order term will control all the negative parts. The term I It is readily to see that the zero order term I = R m (ξ, η, ξ, η). Recalling the curvature assumption (1.3), by the definition of sectional curvature, the term I can be bounded from below:

I ≥ |ξ| 2 |η| 2 -⟨ξ, η⟩ 2 (3.47) = (ξ 1 η 2 -ξ 2 η 1 ) 2 + ξ 2 3 (η 2 1 + η 2 2 ).
The term II The term II involves the first order derivatives of the curvature as coefficients for the terms τ ξ 2 1 η 2 2 , τ ξ 2 2 η 2 1 , τ ξ 1 ξ 2 η 2 2 and τ ξ 2 2 η 1 η 2 . They can not be directly controlled by the related terms of the second order in ξ and η, but the combination of them will compose good terms, more precisely, the term II can be stated as:

II = 3 4 τ ∇ 1 R 1212 ξ 2 2 η 2 1 -τ ∇ 1 R 1212 ξ 1 ξ 2 η 1 η 2 + τ 4 ∇ 1 R 1212 ξ 2 1 η 2 2 + τ 2 ∇ 2 R 1212 (ξ 2 2 η 1 η 2 -ξ 1 ξ 2 η 2 2 ) + 3 2 τ ∇ 1 R 1213 ξ 2 ξ 3 η 2 1 + 3 4 τ ∇ 1 R 1313 ξ 2 3 η + τ [(∇ 1 R 1223 + ∇ 2 R 1213 )ξ 2 ξ 3 + (∇ 1 R 1323 + 1 2 ∇ 2 R 1313 )ξ 2 3 - ∇ 1 R 1213 ξ 1 ξ 3 ]η 1 η 2 + τ [- 1 2 (∇ 1 R 1223 + ∇ 2 R 1213 )ξ 1 ξ 3 + 1 2 ∇ 2 R 1223 ξ 2 ξ 3 + ( 1 4 ∇ 1 R 2323 + 1 2 ∇ 2 R 1323 )ξ 2 3 ]η 2 2 + = τ 4 ∇ 1 R 1212 (ξ 2 η 1 -ξ 1 η 2 ) 2 + τ 2 ∇ 1 R 1212 (ξ 2 η 1 -ξ 1 η 2 )ξ 2 η 1 + τ 2 ∇ 2 R 1212 (ξ 2 η 1 -ξ 1 η 2 )ξ 2 η 2 + 3 2 τ ∇ 1 R 1213 ξ 2 ξ 3 η 2 1 + 3 4 τ ∇ 1 R 1313 ξ 2 3 η 1 + τ [(∇ 1 R 1223 + ∇ 2 R 1213 )ξ 2 ξ 3 + (∇ 1 R 1323 + 1 2 ∇ 2 R 1313 )ξ 2 3 - ∇ 1 R 1213 ξ 1 ξ 3 ]η 1 η 2 + τ [- 1 2 (∇ 1 R 1223 + ∇ 2 R 1213 )ξ 1 ξ 3 + 1 2 ∇ 2 R 1223 ξ 2 ξ 3 + ( 1 4 ∇ 1 R 2323 + 1 2 ∇ 2 R 1323 )ξ 2 3 ]η 2 2 .
Using the curvature assumption (1.4) and the parallel property (1.1), the term II can be estimated as follows:

II ≥ - ε 4 τ (ξ 2 η 1 -ξ 1 η 2 ) 2 - ε 2 τ |(ξ 2 η 1 -ξ 1 η 2 )ξ 2 η 1 | - ε 2 τ |(ξ 2 η 1 -ξ 1 η 2 )ξ 2 η 2 | - 3 2 ετ |ξ 2 ξ 3 η 2 1 | - 3 4 ετ ξ 2 3 η 2 1 - ετ (2|ξ 2 ξ 3 η 1 η 2 | + 3 2 |ξ 2 3 η 1 η 2 | + |ξ 1 ξ 3 η 1 η 2 |) - ετ (|ξ 1 ξ 3 η 2 2 | + 1 2 |ξ 2 ξ 3 η 2 2 | + 3 4 ξ 2 3 η 2 2 ).
In view of the Cauchy Schwartz inequality, it follows that

2τ |(ξ 2 η 1 -ξ 1 η 2 )ξ 2 η 1 | ≤ (ξ 2 η 1 -ξ 1 η 2 ) 2 + τ 2 ξ 2 2 η 2 1 , 2τ |(ξ 2 η 1 -ξ 1 η 2 )ξ 2 η 2 | ≤ (ξ 2 η 1 -ξ 1 η 2 ) 2 + τ 2 ξ 2 2 η 2 2 , 2τ |ξ 2 ξ 3 η 2 1 | ≤ τ 2 ξ 2 2 η 2 1 + ξ 2 3 η 2 1 , 2τ |ξ 2 ξ 3 η 1 η 2 | ≤ τ 2 ξ 2 2 η 2 1 + ξ 2 3 η 2 2 , 2|ξ 2 3 η 1 η 2 | ≤ ξ 2 3 η 2 1 + ξ 2 3 η 2 2 , 2τ |ξ 1 ξ 3 η 1 η 2 | ≤ τ 2 ξ 2 1 η 2 2 + ξ 2 3 η 2 1 , 2τ |ξ 1 ξ 3 η 2 2 | ≤ τ 2 ξ 2 1 η 2 2 + ξ 2 3 η 2 2 , 2τ |ξ 2 ξ 3 η 2 2 | ≤ τ 2 ξ 2 2 η 2 2 + ξ 2 3 η 2 2 .
Finally,

II ≥ -ε( τ 4 + 1 2 )(ξ 1 η 2 -ξ 2 η 1 ) 2 -ετ 2 ξ 2 1 η 2 2 -2ετ 2 ξ 2 2 η 2 1 - 1 2 ετ 2 ξ 2 2 η 2 2 -ε( 5 4 + 3 2 τ )ξ 2 3 η 2 1 -ε( 7 4 + 3 2 τ )ξ 2 3 η 2 2 .
Now assume 0 < τ < 2, the following inequality holds:

II ≥ -ε(ξ 1 η 2 -ξ 2 η 1 ) 2 -ετ 2 ξ 2 1 η 2 2 -2ετ 2 ξ 2 2 η 2 1 - (3.48) 1 2 ετ 2 ξ 2 2 η 2 2 -ε( 5 4 + 3 2 τ )ξ 2 3 η 2 1 -ε( 7 4 + 3 2 τ )ξ 2 3 η 2 2 .
The term III The term III consists of all terms whose coefficients involving the second order derivatives of the curvature. The term ξ 1 ξ 2 η 1 η 2 τ 2 is bad one. It will be handled in the following. The others are all good ones. They are used to control the negative terms in the following.

The term IV We now handle the remainder IV. By definition, there exists a small positive number δ3 which is independent of ε and a positive constant C such that, for any 0 < τ < δ3 ,

IV ≥ -Cτ 2 (ε + τ 2 )(ξ 2 1 η 2 2 + ξ 2 2 η 2 1 + ξ 2 2 η 2 2 + ξ 2 3 η 2 1 + ξ 2 3 η 2 2 + η 2 2 ).
Given a small positive real number ε 1 which will be determined later, assume that 0

< ε < ε3 2C , 0 < τ < min{ √ ε3 2C , δ3 }.
We have:

IV ≥ -ε 3 τ 2 (ξ 2 1 η 2 2 + ξ 2 2 η 2 1 + ξ 2 2 η 2 2 + ξ 2 3 η 2 1 + ξ 2 3 η 2 2 + η 2 2 ). ( 3
C(m, ν)(ξ, η) ≥ (ξ 1 η 2 -ξ 2 η 1 ) 2 + ξ 2 3 (η 2 1 + η 2 2 ) - 1 2 ε 3 (ξ 1 η 2 -ξ 2 η 1 ) 2 - 1 2 ε 3 τ 2 ξ 2 1 η 2 2 -ε 3 τ 2 ξ 2 2 η 2 1 - 1 4 ε 3 τ 2 ξ 2 2 η 2 2 -ε 3 ( 5 8 + 3 4 τ )ξ 2 3 η 2 1 - ε 3 ( 7 8 + 3 4 τ )ξ 2 3 η 2 2 + τ 2 ( 1 15 ξ 2 1 η 2 2 - 2 5 ξ 1 ξ 2 η 1 η 2 + 2 5 ξ 2 2 η 2 1 + 1 15 ξ 2 2 η 2 2 + 2 5 ξ 2 3 η 2 1 + 2 15 ξ 2 3 η 2 2 ) -ε 3 τ 2 (ξ 2 1 η 2 2 + ξ 2 2 η 2 1 + ξ 2 2 η 2 2 + ξ 2 3 η 2 1 + ξ 2 3 η 2 2 + η 2 2 ).
Gathering similar terms, we get

C(m, ν)(ξ, η) ≥ (1 - ε 3 2 )(ξ 1 η 2 -ξ 2 η 1 ) 2 + ( 1 15 - 3 2 ε 3 )τ 2 ξ 2 1 η 2 2 - 2 5 τ 2 ξ 1 ξ 2 η 1 η 2 + ( 2 5 -2ε 3 )τ 2 ξ 2 2 η 2 1 + ( 1 15 - 5 4 ε 3 )τ 2 ξ 2 2 η 2 2 + [ 2 5 τ 2 -ε 3 ( 13 8 + 3 4 τ ) + 1]ξ 2 3 η 2 1 + [ 2 15 τ 2 -ε 3 ( 15 8 + 3 4 τ ) + 1]ξ 2 3 η 2 2 -ε 3 τ 2 η 2 2 = (1 - 3 4 ε 3 )(ξ 1 η 2 -ξ 2 η 1 ) 2 + ( 1 15 -3ε 3 )τ 2 ξ 2 1 η 2 2 - 2 5 τ 2 ξ 1 ξ 2 η 1 η 2 + ( 2 5 -3ε 3 )τ 2 ξ 2 2 η 2 1 + ε 3 4 (ξ 1 η 2 -ξ 2 η 1 ) 2 + 3 2 ε 3 τ 2 ξ 2 1 η 2 2 + ε 3 τ 2 ξ 2 2 η 2 1 + ( 1 15 - 5 4 ε 3 )τ 2 ξ 2 2 η 2 2 + [1 -ε 3 ( 13 8 + 3 4 τ )]ξ 2 3 η 2 1 + [1 -ε 3 ( 15 8 + 3 4 τ )]ξ 2 3 η 2 2 + 2 5 τ 2 ξ 2 3 η 2 1 + 2 15 τ 2 ξ 2 3 η 2 2 -ε 3 τ 2 η 2 = [1 - 3 4 ε 3 + ( 1 15 -3ε 3 )τ 2 ]ξ 2 1 η 2 2 -2(1 - 3 4 ε 3 + τ 2 5 )ξ 1 ξ 2 η 1 η 2 + [1 - 3 4 ε 3 + ( 2 5 -3ε 3 )τ 2 ]ξ 2 2 η 2 1 + ε 3 4 (ξ 1 η 2 -ξ 2 η 1 ) 2 + 3 2 ε 3 τ 2 ξ 2 1 η 2 2 + ε 3 τ 2 ξ 2 2 η 2 1 + ( 1 15 - 5 4 ε 3 )τ 2 ξ 2 2 η 2 2 + [1 -ε 3 ( 13 8 + 3 4 τ )]ξ 2 3 η 2 1 + [1 -ε 3 ( 15 8 + 3 4 τ )]ξ 2 3 η 2 2 + 2 5 τ 2 ξ 2 3 η 2 1 + 2 15 τ 2 ξ 2 3 η 2 2 -ε 3 τ 2 η 2 2 .
Note that the discriminant of the quadratic polynomial

h ε,τ (t) = [1- ε+( 1 15 -3ε)τ 2 ]t 2 - 2(1 -3 4 ε + τ 2 5 )t + 1 -3 4 ε + ( 2 5 -3ε)τ 2 is polynomial with arguments (ε, τ ). It is given by Λ = 4τ 2 [ - 1 15 + 121 20 ε - 9 2 ε 2 + ( 1 75 + 7 5 ε -9ε 2 ) τ 2
] .

Assume ε < 7 45 , there exists a small positive constant ε3 < 1 30 , such that the discriminant is non-positive in [0, ε3 ] × [0, 1], i.e. for any (ε, τ

) ∈ [0, ε3 ] × [0, 1], Λ ≤ 4τ 2 (- 4 75 + 149 20 ε - 27 2 ε 2 ) ≤ 0. So if ε 3 < ε3 , C(m, ν)(ξ, η) ≥ ε 3 4 (ξ 1 η 2 -ξ 2 η 1 ) 2 + [1 -ε 3 ( 13 8 + 3 4 τ )]ξ 2 3 η 1 + [1 -ε 3 ( 15 8 + 3 4 τ )]ξ 2 3 η 2 2 + 3 2 ε 3 τ 2 ξ 2 1 η 2 2 + ε 3 τ ξ 2 2 η 2 1 + ( 1 15 - 5 4 ε 3 )τ 2 ξ 2 2 η 2 2 + 2 5 τ 2 ξ 2 3 η 2 1 + 2 15 τ 2 ξ 2 3 η 2 2 -ε 3 τ 2 η 2 2 .
Assume that

ε 3 < C 2 . For all τ ≤ min{ √ ε3 2C , δ3 }, C(m, ν)(ξ, η) ≥ ε 3 4 (ξ 1 η 2 -ξ 2 η 1 ) 2 + (1 -2ε 3 )ξ 2 3 η 2 1 + (1 - 9 4 ε 3 )ξ 2 3 η 2 2 + 3 2 ε 3 τ 2 ξ 2 1 η 2 2 + ε 3 τ 2 ξ 2 2 η 2 1 + ( 1 15 - 5 4 ε 3 )τ 2 ξ 2 2 η 2 2 + 2 5 τ 2 ξ 2 3 η 2 1 + 2 15 τ 2 ξ 2 3 η 2 2 -ε 3 τ 2 η 2 2 . Replacing η 2 2 by η 2 2 (ξ 2 1 + ξ 2 2 + ξ 2 3 ), we get C(m, ν)(ξ, η) ≥ ε 3 4 (ξ 1 η 2 -ξ 2 η 1 ) 2 + (1 -2ε 3 )ξ 2 3 η 2 1 + (1 - 9 4 ε 3 )ξ 2 3 η 2 2 + 1 2 ε 3 τ 2 ξ 2 1 η 2 2 + ε 3 τ 2 ξ 2 2 η 2 1 + ( 1 15 - 9 4 ε 3 )τ 2 ξ 2 2 η 2 2 + 2 5 τ 2 ξ 2 3 η 2 1 + ( 2 15 -ε 3 )τ 2 ξ 2 3 η 2 2 .
Assume that

ε 3 < min{ 2 135 , C 2 , ε3 }, then for any τ ≤ min{ √ ε3 2C , δ3 }, we have C(m, ν)(ξ, η) ≥ ε 3 4 (ξ 1 η 2 -ξ 2 η 1 ) 2 + 131 135 ξ 2 3 η 2 1 + 29 30 ξ 2 3 η 2 2 + 1 2 ε 3 τ 2 ξ 2 1 η 2 2 + ε 3 τ 2 ξ 2 2 η 2 1 + 1 30 τ 2 ξ 2 2 η 2 2 + 2 5 τ 2 ξ 2 3 η 2 1 + 1 10 τ 2 ξ 2 3 η 2 2 .
We infer the existence of positive constants ε3 , κ 3 and δ 3 such that for ∀ν ∈ T m M with |ν| < δ 3 ,

C(m, ν)(ξ, η) ≥ κ 3 [ (|ξ| 2 |η| 2 -⟨ξ, η⟩ 2 ) + |ν| 2 (ξ 2 2 + ξ 2 3 )|η| 2 + |ν| 2 |ξ| 2 η 2 2
] .

The proof of Theorem 3.2 is thus complete.

Chapter 4

The inverse for the Hessian of the squared distance

This chapter is devoted to the asymptotic property of the inverse of the Hessian of the squared distance. Firstly, we deal with the derivatives of geodesic motion from order one to order three. Secondly, we give a basic formula to calculate the Hessian of the squared distance. Finally, we examine the approximation for the inverse of the Hessian of the squared distance. We begin by some notations.

Let (M, g) be a closed Riemaniann manifold of dimension n ≥ 2. Assume that (M, g) satisfies the curvature assumptions (1.3) and (1.4). Fix m 0 ∈ M, ν 0 ∈ I(m 0 )\{0}. Set γ(t) = exp m0 (tν 0 ). Let {e 1 (t), e 2 (t), • • • , e n (t)} be a parallel orthonormal moving frame of vector fields along the geodesic γ with e 1 (t) = γ(t)

| γ(t)| . Taking the corresponding Fermi coordinate system x along the geodesic exp m0 (t ν0 |ν0| ), the fiber coordinate of

T M → M naturally related to x is denoted by v = (v 1 , v 2 , • • • , v n ).
For each m ∈ M, ν ∈ I(m) with m in the domain of the Fermi coordinate system x, we set:

X = X(x, v, t) = (X 1 (x, v, t), X 2 (x, v, t), • • • , X n (x, v, t)) = x(exp m (tν)), where x = x(m) and ν = v i ∂ i .
Thus, X(x, v, t) is the coordinate of the geodesic exp m tν. Then the n-tuple X = X(x, v, t) is the solution of the Cauchy problem (2.1).

Note that (m 0 , ν 0 ) corresponds to (0, v 0 ) where v 0 = (|ν 0 |, 0) in the Fermi coordinate system x. On the axis, set for short X 0 (t) := X(0, v 0 , t).

The derivatives of the geodesic motion

The first derivatives of the geodesic motion

In this subsection, we are concerned with the first derivatives of the geodesic motion. Recall that X(x, v, t) is the coordinate of the geodesic exp m tν and X = X(x, v, t) is the solution of the Cauchy problem (2.1). It is clear that

Ẋ(0, v 0 , t) = (|ν 0 |t, 0, • • • , 0) T . Let J a be ∂ x X or D v X.
Differentiating (2.1) once with respect to the variable x(or v), on the axis, we obtain the following equation:

Ji a + ∂ l Γ i jk (X) Ẋj Ẋk J l a + 2Γ i jk (X) Ẋj Jk a = 0,
with the initial conditions, namely either

∂ a X i (0) = δ i a , ∂ a Ẋi (0) = 0, 47 or D a X i (0) = 0, D a Ẋi (0) = δ i a .
Evaluating on the axis, and using (1.15), the relations Ẋ(0, v 0 , t) = (|ν 0 |t, 0, • • • , 0) T and (1.12), we get the following equation

Ji a + |ν 0 | 2 R i 1β1 (X 0 )J β a = 0,
This equation is exactly (1.7). Thus from Lemma 1.1, we get Lemma 4.1. [START_REF] Delanoë | Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds[END_REF] On the axis, for t ∈ [0, 1], the terms:

|∂ x X(0, v 0 , t)|, |∂ x Ẋ(0, v 0 , t)|, |D v X(0, v 0 , t)|, |D v Ẋ(0, v 0 , t)|,
are all bounded from above by a positive constant C > 0.

We require the notation ∂ x X 0 (t) and D v X 0 (t) for the solution Ja of the unperturbed equation Ji

a + |ν 0 | 2 Ri 1β1
Jβ a = 0, with the initial conditions, namely either

∂ a X i 0 (0) = δ i a , ∂a X i 0 (0) = 0, or D a X i 0 (0) = 0, Ḋa X i 0 (0) = δ i a .
It is clear that ∂ x X 0 (t) and D v X 0 (t) go back to J1 and J0 respectively on the axis. From Lemma 1.2, we obtain Lemma 4.2. [START_REF] Delanoë | Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds[END_REF] There exists a positive number C > 0 such that on the axis, for all t ∈ [0, 1],, we have

|∂ x X(0, v 0 , t) -∂ x X 0 (t)| ≤ Cε, |∂ x Ẋ(0, v 0 , t) -∂x X 0 (t)| ≤ Cε, |D v X(0, v 0 , t) -D v X 0 (t)| ≤ Cε, |D v Ẋ(0, v 0 , t) -Ḋv X 0 (t)| ≤ Cε.

The second derivatives of the geodesic motion

In this subsection, we handle the second derivatives of geodesic motion. Let J ab be

∂ 2 ab X, ∂ a D b X, D a ∂ b X or D 2 ab X.

Differentiating (2.1) twice with respect to the parameters x and v, we get

Ji ab + ∂ l Γ i jk Ẋj Ẋk J l ab + 2Γ i jk Ẋj Jk ab = -∂ 2 lp Γ i jk Ẋj Ẋk J l a J p b -2∂ l Γ i jk Ẋj ( Jk b J l a + Jk a J l b ) -2Γ i jk Jj b Jk a .
with homogenuous initial conditions:

J i ab (0) = Ji ab (0) = 0.
Evaluating on the axis, and using (1.15), the relations Ẋ(0, v 0 , t) = (|ν 0 |t, 0, • • • , 0) T and (1.12), we obtain

Ji ab + |ν 0 | 2 R i 1α1 (X 0 )J α ab = -|ν 0 | 2 ∂ 2 lp Γ i 11 J l a J p b -2|ν 0 |R i kβ1 (X 0 )( Jk b J β a + Jk a J β b ). (4.1)
Using (1.15), (1.16) and Lemma 4.1, we record a standard result of the second order differential equations, namely:

Lemma 4.3. [START_REF] Delanoë | Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds[END_REF] On the axis, for t ∈ [0, 1], the terms:

|∂ 2 xx X(0, v 0 , t)|, |∂ 2 xx Ẋ(0, v 0 , t)|, |∂ x D v X(0, v 0 , t)|, |∂ x D v Ẋ(0, v 0 , t)|, |D 2 vv X(m, v 0 , t)|, |D 2 vv Ẋ(0, v 0 , t)|
are all bounded from above by a positive constant C > 0.

Let us introduce the solutions

∂ 2 xx X 0 , ∂ x D v X 0 , D v ∂ x X 0 and D 2
vv X 0 along the axis of the unperturbed equation:

Ji ab + |ν 0 | 2 δ i α Jα ab = -2|ν 0 |(δ i β δ 1 k -δ i 1 δ β k )( Jβ a Jk b + Jβ b Jk a ), (4.2)
with homogenuous initial conditions Ji ab (0) = Ji ab (0) = 0. (4.3) Lemma 4.4. [START_REF] Delanoë | Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds[END_REF] There exists a positive number C > 0 such that on the axis, for all t ∈ [0, 1], we have The last two approximations are just the consequences of the facts ∂ 2 1x X 0 (t) = 0 and ∂ 1 D v X 0 (t) = 0 which follow from the equation (4.2) and the initial condition (4.3).

|∂ 2 xx X(0, v 0 , t) -∂ 2 xx X 0 (t)| ≤ Cε, |∂ 2 xx Ẋ(0, v 0 , t) - ∂2 xx X 0 (t)| ≤ Cε, |∂ x D v X(0, v 0 , t) -∂ x D v X 0 (t)| ≤ Cε, |∂ x D v Ẋ(0, v 0 , t) - ∂x D v X 0 (t)| ≤ Cε, |D 2 vv X(0, v 0 , t) -D 2 vv X 0 (t)| ≤ Cε, |D 2 vv Ẋ(0, v 0 , t) - Ḋ2 vv X 0 (t)| ≤ Cε. Furthermore,|∂ 2 1x X(0, v 0 , t)| ≤ Cε, |∂ 1 D v X(0, v 0 , t)| ≤ Cε. Proof.

The third derivatives of the geodesic motion

In this section, we address the third derivatives of the geodesic motion. Let

J i abc (t) be ∂ 3 xxx X(0, v 0 , t), ∂ 2 xx D v X(0, v 0 , t), ∂ x D 2 vv X(0, v 0 , t) or D 3
vvv X(0, v 0 , t). Differentiating (2.1) three times with respect to the variable x and v :

Ji abc + |ν 0 | 2 R i 1α1 (X 0 )J α abc = -|ν 0 | 2 ∂ 3 lpq Γ i 11 J l a J p b J q c -|ν 0 | 2 ∂ 2 lp Γ i 11 ∑ (a,b,c) J l ab J p c -2|ν 0 |∂ 2 lp Γ i 1k ∑ (a,b,c) Jk a J l b J p c -2|ν 0 |R i kβ1 (X 0 ) ∑ (a,b,c) ( Jk a J β bc + Jk ab J β c ) -2∂ β Γ i jk ∑ (a,b,c) Jj a Jk b J β c .
with homogenuous initial conditions

J i abc (0) = J i abc (0) = 0.
Repeating the procedure in Lemma 4.3 we get: Lemma 4.5. [START_REF] Delanoë | Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds[END_REF] On the axis, for t ∈ [0, 1], the terms:

|∂ 3 xxx X(0, v 0 , t)|, |∂ 3 xxx Ẋ(0, v 0 , t)|, |∂ 2 xx D v X(0, v 0 , t)|, |∂ 2 xx D v Ẋ(0, v 0 , t)|, |∂ x D 2 vv X(0, v 0 , t)|, |∂ x D 2 vv Ẋ(0, v 0 , t)|, |D 3 vvv X(0, v 0 , t)|, |D 3 vvv Ẋ(0, v 0 , t)|
are bounded from above by a positive constant C > 0 (independent of t, v 0 ).

Let us introduce the solutions

∂ 3 xxx X 0 , ∂ 2 xx D v X 0 , ∂ x D v ∂ x X 0 , D v ∂ 2 xx X 0 , ∂ x D 2 vv X 0 , D v ∂ x D v X 0 , D 2
vv ∂ x X 0 and D 3 vvv X 0 along the axis of the unperturbed equation:

Ji abc + |ν| 2 δ i α Jα abc (4.4) = 4 3 (δ i k -δ i 1 δ 1 k ) ∑ (a,b,c) (|ν| 2 Jk a Jβ b Jβ c -2 Jβ a Jβ b Jk c ) - 2|ν|(δ i β δ 1 k -δ i 1 δ β k ) ∑ (a,b,c) ( Jk a Jβ bc + Jk ab Jβ c ) - 2(δ i k -δ i 1 δ 1 k ) ∑ (a,b,c) J1 a J1 b Jk c + [2δ i 1 δ 1 k + 2 3 (δ i k -δ i 1 δ 1 k )] ∑ (a,b,c) ( Jk a Jβ b Jβ c + Jβ a Jk b Jβ c ).
with homogenuous initial conditions Ji abc (0) = Ji abc (0) = 0.

Repeating the procedure in Lemma 4.4, we get Lemma 4.6. [START_REF] Delanoë | Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds[END_REF] There exists a positive number C > 0 such that on the axis, for all t ∈ [0, 1], we have

|∂ 3 xxx X(0, v 0 , t) -∂ 3 xxx X 0 (t)| ≤ Cε, |∂ 2 xx D v X(0, v 0 , t) -∂ 2 xx D v X 0 (t)| ≤ Cε, |∂ x D 2 vv X(0, v 0 , t) -∂ x D 2 vv X 0 (t)| ≤ Cε, |D 3 vvv X(0, v 0 , t) -D 3 vvv X 0 (t)| ≤ Cε.

The Hessian of the squared distance

In this section, we compute the local expression of the Hessian of the squared distance. We start from the well-known identity(p.156 [START_REF] Jost | Riemannian Geometry and Geometric Analysis[END_REF]):

p 2 = exp p1 [-grad p1 c(•, p 2 )], (4.5)
where the identity makes sense whenever (p 1 , p 2 ) ∈ M × M are no cut points of each other. Suppose that the points p 1 and p 2 lie in the domain of the Fermi coordinate system x. The coordinates of p 1 and p 2 are given by x 1 = x(p 1 ) and x 2 = x(p 2 ) respectively. Let m be in the domain of the Fermi coordinate system x. Set ν ∈ I(m). The coordinate of ν is denoted by v = v i ∂ i i.e.v i = dx i (ν). Differentiating (4.5) with respect to the coordinates x 1 at x 1 = x(m), we get for X(x 1 , v, t) at x 1 = x(m), t = 1 and at v = v i ∂ i given by exp m ν = p 2 , the following identity:

D k X i (x, v, 1)∇ k j c(m, exp m ν) = δ ∇ j X i (x, v, 1), (4.6) where δ ∇ j X i = ∂ j X i -Γ p jl (x)v l D p X i
. This is the fundamental formula to compute the approximation of the inverse for the Hessian of the squared distance.

The inverse for the Hessian of the squared distance

In the last section, we consider the approximation of the inverse of Hessian of the squared distance. The result is presented as follows.

Theorem 4.1. Let (M, g) be a closed n-dimensional Riemannian manifold satisfying (1.3) and (1.4). Then there exists positive numbers C > 0 and ε 0 > 0 depending only n such that, for any ε < ε 0 and for any

m 0 ∈ M, ν 0 ∈ I(m 0 ), |ν 0 | m0 ≥ 3π
4 , the following inequalities hold

1) |S -1 (m 0 , ν 0 , 1) -S-1 (m 0 , ν 0 , 1)| ≤ Cε; 2) |∂ x S -1 (m 0 , ν 0 , 1) -∂ x S-1 (m 0 , ν 0 , 1)| ≤ Cε, |D v S -1 (m 0 , ν 0 , 1) -D v S-1 (m 0 , ν 0 , 1)| ≤ Cε; 3) |∂ 2 xx S -1 (m 0 , ν 0 , 1) -∂ 2 xx S-1 (m 0 , ν 0 , 1)| ≤ Cε, |∂ x D v S -1 (m 0 , ν 0 , 1) -∂ x D v S-1 (m 0 , ν 0 , 1)| ≤ Cε, |D 2 vv S -1 (m 0 , ν 0 , 1) -D 2 vv S-1 (m 0 , ν 0 , 1)| ≤ Cε. Proof. Set m 0 ∈ M, ν 0 ∈ I(m 0 ), |ν 0 | m0 ≥ 3π 4 .
Let x be the Fermi coordinate system along the geodesic exp m0 (t ν0 |ν0| ) and v be the fiber coordinate of T M → M naturally associated to x. Set r 0 = |ν 0 |. 1) Making use of (b) in Proposition 1.1, we know that the matrix of the linear operator S(m 0 , ν 0 , 1) from T m M to T m M in the orthonormal basis {e 1 (0), e 2 (0), • • • , e n (0)} is given by J -1 0 J 1 . As a consequence, the matrix to the inverse of S is given by J -1 1 J 0 . It is clear that 1) is just a result of Lemma 4.1 and Lemma 4.2.

2) The inverse of S is denoted by A. The components of A in the Fermi coordinate system x is denoted by A i j . From the formula (4.6) and (b) in Proposition 1.1, we obtain the formula

A i j (x, v) = Z i k D j X k , ( 4.7)
where

Z i k δ ∇ j X k = δ i j .
Differentiating (4.7) once with respect to x :

∂ a A i j = -Z i p ∂ a δ ∇ q X p Z q k D j X k + Z i k ∂ a D j X k = -Z i p [∂ 2 aq X p -∂ a Γ d ql (x)v l D d X p - Γ d ql (x)v l ∂ a D d X p ]Z q k D j X k + Z i k ∂ a D j X k .
Evaluating on the axis, we have

∂ a A i j = -Z i p (∂ 2 aq X p -r 0 R d qa1 D d X p )Z q k D j X k + Z i k ∂ a D j X k .
Applying Lemma 4.1, Lemma 4.2, Lemma 4.3 and Lemma 4.4, and together with the facts ∂ 2 aj X i 0 (t) = (-r 0 t + sin r 0 t cos r 0 t)δ i 1 (δ ajδ 1a δ 1j ).

∂ a D j X i 0 (t) = -δ 1j (δ i a -δ i 1 δ 1a )t sin r 0 t + sin 2 r 0 t r 0 δ i 1 (δ aj -δ 1a δ 1j ), we get ∂ a A i j = B(ε).
To calculate the corresponding derivatives on the sphere, we need to give the formula of S-1 . From the Remark 1.1(2), we see that

S(m, ν, 1)(ξ) = ξ -(1 -|ν| cot(|ν|))(ξ -⟨ξ, ν |ν| ⟩ ν |ν| ).
It can be checked that the inverse of S which is denoted by Ā takes the form

Ā(ξ) = ξ -(1 - |ν| tan |ν| )(ξ -⟨ξ, ν |ν| ⟩ ν |ν| ). Differentiating Āi j = δ i j -(1 -|ν| tan |ν| )(δ i j - g jk (x)v i v k |ν| 2
) with respect to x and evaluating at (m, ν), we get ∂ a Āi j = B(ε). Differentiating (4.7) once with respect to v, we infer

D a A i j = -Z i p D a δ ∇ q X p Z q k D j X k + Z i k D 2 aj X k = -Z i p [D a ∂ q X p -Γ d qa (x)D d X p - Γ d ql (x)v l D 2 ad X p ]Z q k D j X k + Z i k D 2 aj X k

Evaluating on the axis, this gives

D a A i j = -Z i p D a ∂ q X p Z q k D j X k + Z i k D 2 aj X k Applying Lemma 4.1, Lemma 4.
D a ∂ j X i 0 (t) = ∂ j D a X i 0 (t). D 2 aj X i 0 (t) = ( 1 r 0 t cos r 0 t - 1 r 2 0 sin r 0 t)(δ i a δ 1j + δ i 1 δ aj -2δ i 1 δ 1a δ 1j ) + ( t r 0 - 1 r 2 0 sin r 0 t cos r 0 t)δ 1a (δ i j -δ i 1 δ 1j ),
we obtain

D a A i j = ( 1 r 0 - tan r 0 r 2 0 )(δ i a δ 1j + δ i 1 δ aj -2δ i 1 δ 1a δ 1j ) + ( sec 2 r 0 r 0 - tan r 0 r 2 0 )δ 1a (δ i j -δ i 1 δ 1j ) + B(ε) = D a Āi j + B(ε).
3)Differentiating (4.7) twice with respect to x, we have

∂ 2 ab A i j = Z i c ∂ a δ ∇ d X c Z d p ∂ b δ ∇ q X p Z q k D j X k + (4.8) Z i c ∂ b δ ∇ d X c Z d p ∂ a δ ∇ q X p Z q k D j X k - Z i p ∂ 2 ab δ ∇ q X p Z q k D j X k -Z i p ∂ a δ ∇ q X p Z q k ∂ b D j X k - Z i p ∂ b δ ∇ q X p Z q k ∂ a D j X k + Z i k ∂ 2 ab D j X k Using Lemma 4.1, Lemma 4.
∂ 3 abj X i 0 (t) = ( r 0 2 t sin r 0 t + 1 6 sin 2 r 0 t cos r 0 t)(δ i k -δ i 1 δ 1 k ) ∑ (a,b,j) δ k a δ β b δ β j + 1 2 (r 0 t sin r 0 t -sin 2 r 0 t cos r 0 t)(δ i k -δ i 1 δ 1 k ) ∑ (a,b,j) δ k a (δ bj -δ 1 b δ 1 j ) ∂ 2 ab D j X i 0 (t) = ( sin 2 r 0 t 6r 0 - t cos r 0 t 2 + sin r 0 t 2r 0 )(δ i k -δ i 1 δ 1 k )(δ k a δ β b δ β j + δ β a δ k b δ β c ) + [( sin 2 r 0 t 6r 0 + t cos r 0 t 2 - sin r 0 t 2r 0 )(δ i k -δ i 1 δ 1 k ) -2δ i 1 δ 1 k t sin r 0 t]δ β a δ β b δ k j + 1 2 (- sin 2 r 0 t r 0 + t cos r 0 t - sin r 0 t r 0 )(δ i k -δ i 1 δ 1 k )[δ k a (δ bj -δ 1 b δ 1 j ) + δ k b (δ aj -δ 1 a δ 1 j )] + (- sin 2 r 0 t 2r 0 - 3t cos r 0 t 2 + 3 sin r 0 t 2r 0 )(δ i k -δ i 1 δ 1 k )δ k j (δ ab -δ 1 a δ 1 b ),
we obtain

∂ 2 ab A i j = ( tan r 0 r 0 -sec 2 r 0 )(δ ab -δ 1a δ 1b )(δ i j -δ i 1 δ 1j ) + B(ε) = ∂ 2 ab Āi j + B(ε).
Differentiating (4.7) with respect to x and v respectively, we infer

∂ a D b A i j = Z i c ∂ a δ ∇ d X c Z d p D b δ ∇ q X p Z q k D j X k + (4.9) Z i c D b δ ∇ d X c Z d p ∂ a δ ∇ q X p Z q k D j X k - Z i p ∂ a D b δ ∇ q X p Z q k D j X k -Z i p ∂ a δ ∇ q X p Z q k D 2 bj X k - Z i p D b δ ∇ q X p Z q k ∂ a D j X k + Z i k ∂ a D 2 bj X k
In view of Lemma 4.1, Lemma 4.2, Lemma 4.3 and Lemma 4.4,Lemma 4.5, Lemma 4.6, and using the facts

∂ a D b ∂ j X i 0 (t) = ∂ 2 aj D b X i 0 (t). ∂ a D 2 bj X i 0 (t) = (- t sin r 0 t 2r 0 - sin 2 r 0 t cos r 0 t 6r 2 0 )(δ i k -δ i 1 δ 1 k )δ k a δ β b δ β j + [( t sin r 0 t 2r 0 - sin 2 r 0 t cos r 0 t 6r 2 0 )(δ i k -δ i 1 δ 1 k ) + ( 2t r 0 sin r 0 t cos r 0 t - sin 2 r 0 t r 2 0 )δ i 1 δ 1 k ](δ β a δ k b δ β j + δ β a δ β b δ k j ) + (- t sin r 0 t 2r 0 + 1 2r 2 0 sin 2 r 0 t cos r 0 t)(δ i k -δ i 1 δ 1 k ) ∑ (a,b,j) δ k a (δ bj -δ 1 b δ 1 j ) + -t 2 cos r 0 t(δ i k -δ i 1 δ 1 k )δ k a δ 1 b δ 1 j ,
we have

∂ a D b A i j = B(ε)
Similarly, differentiating (4.7) twice with respect to v, we obtain

D 2 ab A i j = Z i c D a δ ∇ d X c Z d p D b δ ∇ q X p Z q k D j X k + (4.10) Z i c D b δ ∇ d X c Z d p D a δ ∇ q X p Z q k D j X k - Z i p D 2 ab δ ∇ q X p Z q k D j X k -Z i p D a δ ∇ q X p Z q k D 2 bj X k - Z i p D b δ ∇ q X p Z q k D 2 aj X k + Z i k D 3 abj X k
Thanks of Lemma 4.1, Lemma 4.2, Lemma 4.3 and Lemma 4.4,Lemma 4.5, Lemma 4.6, together with the facts

D 2 ab ∂ j X i 0 (t) = ∂ j D 2 ab X i 0 (t). ∂ a D 2 bj X i 0 (t) = [(- sin 3 r 0 t 6r 3 0 - t cos r 0 t 2r 2 0 + sin r 0 t 2r 3 0 )(δ i k -δ i 1 δ 1 k ) + (- 2 r 2 0 t cos 2 r 0 t + 2 r 3 0 sin r 0 t cos r 0 t)δ i 1 δ 1 k ] ∑ (a,b,j) δ k a δ β b δ β j + ( sin 3 r 0 t 2r 3 0 + 3t cos r 0 t 2r 2 0 - 3 sin r 0 t 2r 3 0 )(δ i k -δ i 1 δ 1 k ) ∑ (a,b,j) δ k a (δ bj -δ 1 b δ 1 j ) + (- t 2 r 0 sin r 0 t + 2 r 3 0 sin r 0 t - 2t cos r 0 t r 2 0 )(δ i k -δ i 1 δ 1 k ) ∑ (a,b,j) δ k a δ 1 b δ 1 j ,
we get

D 2 ab A i j = D 2 ab Āi j + B(ε).
This finishes the proof of Theorem 4.1.

Delanoë(see Proposition 4.1 in [START_REF] Delanoë | On the smoothness of the potential function in Riemannian optimal transport[END_REF]) reduced the closedness to an uniform upper bound on the Hessian of the classical solutions u t for all t ∈ I. Moreover, Delanoë(see [START_REF] Delanoë | On the smoothness of the potential function in Riemannian optimal transport[END_REF]p.50) also showed that the existence of an uniform upper bound on the Hessian of the classical solutions u t is equivalent to the following two estimates:

(1) There exists a positive constant δ 0 , such that

∀(t, m) ∈ I × M, det(d ∇mut exp m ) ≥ δ 0 , ( 5.2)
(2) There exists a positive constant C, such that

Hess m u t + S(m, ∇ m u t , 1) ≤ CId m , (5.3)
for any (t, m) ∈ I × M.

In conclusion, Delanoë(see [START_REF] Delanoë | On the smoothness of the potential function in Riemannian optimal transport[END_REF]) derived the following fact.

Lemma 5.1. Given a closed Riemannian manifold and given (k, α) ∈ N × (0, 1), with k ≥ 2, the potential function u is C k+2,α for every couple of C k,α positive probability measures (ρ 0 dvol, ρ 1 dvol), if, for each such couple, the requirements (5.2) and ( 5.3) are fulfilled. Moreover, if either ( 1) or ( 2) fails, there exists a number t 0 ∈ [0, 1] such that, the potential function

u t0 is not C 2 .
The first result of this chapter shows the regularity of the optimal transport map on nearly spherical manifold. Theorem 5.1. Let (M, g) be a closed n-dimensional Riemannian manifold satisfying the curvature assumption (1.3). Then there exists a positive constant ε 0 depending only

n such that if ∥Riem - 1 2 g g∥ C 2 (M,g) < ε 0 ,
then for any couple (k, α) ∈ N × (0, 1), with k ≥ 2, the potential function of the optimal transport map is C k+2,α for every couple (ρ 0 dvol, ρ 1 dvol) of C k,α positive Borel probability measures on M.

A direct result of Theorem 5.1 is the smoothness of the optimal transport maps on nearly spherical manifold. Corallary 5.1. Under the same assumptions as in Theorem 5.1, let ρ 0 dvol and ρ 1 dvol be two smooth positive Borel probability measures on M. Then the optimal transport map is smooth.

The second result of this chapter concentrates on the regularity of the optimal transport map on the product of nearly spherical manifold. Theorem 5.2. Let M 1 and M 2 be two closed Riemannian manifolds of dimension n 1 ≥ 2 and n 2 ≥ 2 respectively. Suppose that both M 1 and M 2 satisfy the assumptions (1.3). There exists some positive constant ε 0 > 0 such that if (1.4) holds on M 1 and M 2 with ε < ε 0 , then for any couple (k, α) ∈ N × (0, 1), with k ≥ 2, the potential function of the optimal transport map is C k+2,α for every couple (ρ 0 dvol, ρ 1 dvol) of C k,α positive Borel probability measures on the Riemannian product

M 1 × M 2 .
As a direct consequence of Theorem 5.2, we get the smoothness of the optimal transport maps on the product of nearly spherical manifolds. Corallary 5.2. Under the same hypothesis as in Theorem 5.2, let ρ 0 dvol and ρ 1 dvol be two smooth positive Borel probability measures on M 1 × M 2 . Then the corresponding optimal transport map is smooth.

At the end of this chapter, we derive that the optimal transport map may not be smooth on some manifolds sufficiently close to the product of the standard spheres in C 4 norm. More generally, we prove the following result. Theorem 5.3. Let (M 1 , g 1 ) and (M 2 , g 2 ) be two closed Riemannian manifolds of dimension n 1 ≥ 2 and n 2 ≥ 2 respectively. The product metric produced by g 1 and g 2 is denoted by g × . Then, for any given ε > 0, there exists a metric g on M 1 × M 2 that is conformal to g × and satisfies ∥gg × ∥ C 4 < ε, such that there exist ρ 0 dvol and ρ 1 dvol two smooth positive Borel probability measures on M 1 × M 2 , the corresponding optimal transport map on (M 1 × M 2 , g) is not smooth.

If (M 1 , g 1 ) and (M 2 , g 2 ) satisfy the conditions of Theorem 5.1, we know that the optimal transport map is smooth on (M 1 × M 2 , g × ). By Theorem 5.3, we also know that the smoothness of the optimal transport map is not stable on the perturbed metric of (M 1 × M 2 , g × ).

The smoothness of the optimal transport map on

nearly spherical manifold

Preliminary

In this subsection, we will establish some a priori estimates of the Monge-Ampère type equation (5.1) and a key proposition. As a consequence, the oscillation of u is bounded above by D 2 . The vanishing average gives the results.

As mentioned in section 5.1, the existence of a uniform upper bound on the Hessian of the classical solutions u t is equivalent to estimates (5.2) and (5.3). The Ma-Trudinger-Wang's estimate reduces the existence of a uniform upper bound on the Hessian of the classical solutions u t to the estimate(5.2) under the assumption that the MTW tensor satisfies the A3S condition. Lemma 5.3. Let (M, g) be a closed Riemannian manifold of dimension n ≥ 2. Assume that the MTW tensor of M satisfies the A3S condition. Let ρ 0 dvol and ρ 1 dvol be C 2 positive Borel probability measures on M. If the requirement (5.2) is fulfilled, then there exists a positive constant C depending on n, δ, max I×M |d log ρ t | C 2 , such that the norm of Hess (c) m u t is bounded from above C, for every couple (t, m) ∈ I × M.

Proof. The proof is given in Delanoë [START_REF] Delanoë | On the smoothness of the potential function in Riemannian optimal transport[END_REF].

To conclude this subsection, we give a key proposition which is used in next subsection.

Let J (m, ν) = -|ν| 2 S -1 (m, ν, 1). We are interested in the behaviour of J when |ν| ≥ 3π 4 . So we assume that |ν| ≥ 3π 4 in the following. It is clear that -|ν| 2 is the first eigenvalue of J with the associated eigenvector ν |ν| . By virtue of 1) in Theorem 4.1, there exists a positive number C > 0 such that

|J (m, ν) -J (m, ν)| ≤ Cε.
(

Let us now describe the behavior for the derivatives of J .

Proposition 5.1. Let (M, g) be a closed n-dimensional Riemannian manifold satisfying the curvature assumptions (1.3) and (1.4).Then there exist positive constants ε 0 and C > 0 such that, for any ε < ε 0 , and for any

m 0 ∈ M, ν 0 ∈ I(m 0 ), |ν 0 | ≥ π -δ, 0 < δ < π 4
, the absolute value of the first and second partial derivatives of the components J i j with respect to (x, v) at the point (0, |ν 0 |, 0), are all bounded from above by Cε, except the following partial derivatives:

D 1 J i i , D β J 1 β = D β J β 1 , ∂ 2 ββ J i i , D 2 11 J i i , D 2 1β J β 1 = D 2 β1 J β 1 = D 2 1β J 1 β = D 2 β1 J 1 β , D 2 ββ J i i , D 2 µκ J µ κ = D 2 κµ J µ κ = D 2 µκ J κ µ = D 2 κµ J κ µ , µ ̸ = κ.
and the following estimates hold:

|D 1 J 1 1 + 2π| ≤ C(ε + δ), |D 1 J α α + π| ≤ C(ε + δ), (5.5
)

|D β J β 1 + π| ≤ C(ε + δ), (5.6) |∂ 2 ββ J 1 1 -2π 2 | ≤ C(ε + δ), |∂ 2 ββ J α α -π 2 | ≤ C(ε + δ), (5.7)
|D 2 11 J i i + 2| ≤ C(ε + δ), |D 2 1β J β 1 | ≤ C(ε + δ), (5.8
)

|D 2 ββ J 1 1 | ≤ C(ε + δ), |D 2 ββ J α α + 1 + 2δ α β | ≤ C(ε + δ), (5.9
)

|D 2 µκ J µ κ + 1| ≤ C(ε + δ), µ ̸ = κ.
(5.10)

Proof. Let x be the Fermi coordinate system associated to the geodesic exp m0 s ν0 |ν0| and v be the fiber coordinates of T M → M naturally associated to x.

Set g = g ij (x)dx i dx j , ν = v i ∂x i , J = -|ν| 2 S-1 (m, ν, 1)
. The components of J in the Fermi coordinate system are denoted by J i j , i.e. J = J i j dx j ⊗ ∂ ∂x i . In view of Theorem 4.1, there exists a positive constant C > 0 such that

|∂ x J (m 0 , ν 0 ) -∂ x J (m 0 , ν 0 )| ≤ Cε, (5.11) |D v J (m 0 , ν 0 ) -D v J (m 0 , ν 0 )| ≤ Cε, (5.12) |∂ 2 xx J (m 0 , ν 0 ) -∂ 2 xx J (m 0 , ν 0 )| ≤ Cε, (5.13) |∂ x D v J (m 0 , ν 0 ) -∂ x D v J (m 0 , ν 0 )| ≤ Cε, (5.14) |D 2 vv J (m 0 , ν 0 ) -D 2 vv J (m 0 , ν 0 )| ≤ Cε. ( 5.15)
Thus we only need to calculate the following derivatives:

∂ a J i j , D a J i j , ∂ 2 ab J i j , ∂ a D b J i j , D 2 ab J i j .
Let us compute the above derivatives of J . To differentiate the components of J i j , we need an explicit formula for J i j . By virtue of ( 7), the map J has the expression

J (ξ) = -|ν| 2 S-1 (ξ) = -|ν| 2 ξ + (|ν| 2 -|ν| tan |ν|)(ξ -⟨ξ, ν |ν| ⟩ ν |ν| ).
As a consequence, the components of J in the Fermi coordinate system x are given by

J i j = -|ν| 2 δ i j + (|ν| 2 -|ν| tan |ν|)(δ i j -g jk (x) v i |ν| v k |ν| ) = -|ν| 2 δ i j + φ(|ν|)(|ν| 2 δ i j -g jk v i v k ). (5.16)
where φ(|ν|) = 1 -tan |ν| |ν| . Let us compute the first order derivatives of J i j . Differentiating (5.16) with respect to x and v respectively, one has

∂ a J i j = -∂ a g kl v k v l δ i j + φ 2|ν| ∂ a g pq v p v q (|ν| 2 δ i j -g jk v i v k ) + φ(∂ a g kl v k v l δ i j -∂ a g jk v i v k ), D a J i j = -2g ak v k δ i j + φ |ν| g ap v p (|ν| 2 δ i j -g jk v i v k ) + φ(2g ak v k δ i j -g jk δ i a v k -g ja v i ).
Set r 0 = |ν 0 |. As ∂ i g kl = 0 at the point m 0 , we have at the point (m 0 , ν 0 ) = (0, r 0 , 0),

∂ a J i j = 0.
Combining with (5.11), there holds that

|∂ a J | ≤ Cε.
Recalling g ij = δ ij at the point m 0 , we obtain

D a J i j = -2r 0 δ 1 a δ i j + r 2 0 φδ 1 a (δ i j -δ i 1 δ 1 j ) + r 0 φ(2δ 1 a δ i j -δ i a δ 1 j -δ i 1 δ a j ) = -2r 0 δ 1 a δ i 1 δ 1 j + (r 2 0 φ -2r 0 )δ 1 a (δ i j -δ i 1 δ 1 j ) + r 0 φ(2δ 1 a δ i j -δ i a δ 1 j -δ i 1 δ a j ) = -2r 0 δ 1 a δ i 1 δ 1 j + (tan r 0 -r 0 sec 2 r 0 -2r 0 )δ 1 a (δ i j -δ i 1 δ 1 j ) + (r 0 -tan r 0 )(2δ 1 a δ i j -δ i a δ 1 j -δ i 1 δ a j ). If a = 1, D 1 J i j = -2r 0 δ i 1 δ 1 j + (tan r 0 -r 0 sec 2 r 0 -2r 0 )(δ i j -δ i 1 δ 1 j ) + 2(r 0 -tan r 0 )(δ i j -δ i 1 δ 1 j ) = -2r 0 δ i 1 δ 1 j + (-tan r 0 -r 0 sec 2 r 0 )(δ i j -δ i 1 δ 1 j ). Observe that D 1 J i j = 0 if i ̸ = j.
Thus we only consider the case i = j.

D 1 J i i = -2r 0 δ i 1 + (-tan r 0 -r 0 sec 2 r 0 )(1 -δ i 1 ) = -2πδ i 1 -π(1 -δ i 1 ) + B(δ).
Together with (5.12), this gives (5.5).

If a > 1 (we note by β)

D β J i j = (-r 0 + tan r 0 )(δ i β δ 1 j + δ i 1 δ β j ).
In particular, D β J i j ̸ = 0 except when i = 1, j = β or i = β, j = 1. By symmetry of J , it remains to prove the case i = β, j = 1.

D β J β 1 = -r 0 + tan r 0 = -π + B(δ).
Now by (5.12), we get (5.6).

Let us compute the second order derivatives of J i j . Differentiating (5.16) twice with respect to x, one has

∂ 2 ab J i j = -∂ 2 ab g kl v k v l δ i j + [( φ 2|ν| 2 - φ 2|ν| 3 )∂ a g lh v l v h ∂ b g pq v p v q + φ 2|ν| ∂ 2 ab g pq v p v q ](|ν| 2 δ i j -g jk v i v k ) + φ 2|ν| ∂ b g pq v p v q (∂ a g kl v k v l δ i j -∂ a g jk v i v k ) + φ 2|ν| ∂ a g pq v p v q (∂ b g kl v k v l δ i j -∂ b g jk v i v k ) + φ(∂ 2 ab g kl v k v l δ i j -∂ 2 ab g jk v i v k ).
Evaluating at the point (m 0 , ν 0 ) = (0, r 0 , 0), and by virtue of (1.12), we have

∂ 2 ab J i j = -r 2 0 ∂ 2 ab g 11 δ i j + r 3 0 φ 2 ∂ 2 ab g 11 (δ i j -δ i 1 δ 1 j ) + r 2 0 φ(∂ 2 ab g 11 δ i j -∂ 2 ab g j1 δ i 1 ). If a = 1, 1 ≤ b ≤ n or 1 ≤ a ≤ n, b = 1, since ∂ 2 1i g kl = 0 at the point (m 0 , ν 0 ), we thus obtain ∂ 2 1b J i j = 0 or ∂ 2 a1 J i j = 0. From (5.13), we derive |∂ 2 1b J | ≤ Cε or |∂ 2 a1 J | ≤ Cε. If a, b > 1, for j = 1, ∂ 2 ab J i 1 = -r 2 0 ∂ 2 ab g 11 δ i 1 .
Together with (1.4) and the first expression in (1.13), it follows that

∂ 2 ab J i 1 = 2r 2 0 δ ab δ i 1 + B(ε).
As a consequence, ∂ 2 ab

J i 1 = B(ε) if i > 1 or a ̸ = b, thus |∂ 2 ab J i 1 | ≤ Cε. It suffices to consider the case i = 1 and a = b. ∂ 2 ββ J 1 1 = 2r 2 0 + B(ε) = 2π 2 + B(ε + δ).
Making use of (5.13), we infer the first inequality in (5.7). For j > 1, by the symmetry of J , we assume that i > 1. Recalling (1.13),(1.4), we derive

∂ 2 ab J i j = 2r 2 0 δ ab δ i j -r 3 0 φδ ab δ i j -2r 2 0 φδ ab δ i j + B(ε) = (2r 2 0 -2r 2 0 φ -r 3 0 φ)δ ab δ i j + B(ε) = (r 0 tan r 0 + r 2 0 sec 2 r 0 )δ ab δ i j + B(ε). Note that ∂ 2 ab J i j = B(ε) if a ̸ = b or i ̸ = j, thus it suffices to consider the case a = b and i = j. ∂ 2 ββ J α α = r 0 tan r 0 + r 2 0 sec 2 r 0 + B(ε) = π 2 + B(ε + δ).
Exploiting (5.13), we infer the second expression in (5.7).

Differentiating (5.16) twice with respect to x and v respectively, one has

∂ a D b J i j = -2∂ a g bk v k δ i j + [( φ |ν| 2 - φ |ν| 3 )∂ a g lh v l v h g bp v p + φ |ν| ∂ a g bp v p ](|ν| 2 δ i j -g jk v i v k ) + φ |ν| g bp v p (2∂ a g kl v k v l δ i j -∂ a g jk v i v k ) + φ 2|ν| ∂ a g pq v p v q (2g bk v k δ i j -g jk δ i a v k -g jb v i ) + φ(2∂ a g bk v k δ i j -∂ a g jk δ i b v k -∂ a g jb v i ).
Evaluating at the point (m 0 , ν 0 ) = (0, r 0 , 0), and together with the fact ∂ i g kl = 0 at the point m 0 , this yields

∂ a D b J i j = 0. Combining with (5.11), we get |∂ a D b J | ≤ Cε.
Differentiating (5.16) twice with respect to v, one has

D 2 ab J i j = -2g ab δ i j + [( φ |ν| 2 - φ |ν| 3 )g ap v p g bq v q + φ |ν| g ab ](|ν| 2 δ i j -g jk v i v k ) + φ |ν| g bp v p (2g ak v k δ i j -g jk δ i a v k -g ja v i ) + φ |ν| g ap v p (2g bk v k δ i j -g jk δ i b v k -g jb v i ) + φ(2g ab δ i j -g ja δ i b -g jb δ i a ).
Evaluating at the point (m 0 , ν 0 ), and using the fact g ij = δ ij at the point m 0 , there holds

D 2 ab J i j = -2δ ab δ i j + [(r 2 0 φ -r 0 φ)δ 1 a δ 1 b + r 0 φδ ab ](δ i j -δ i 1 δ 1 j ) + r 0 φδ 1 b (2δ 1 a δ i j -δ i a δ 1 j -δ i 1 δ a j ) + r 0 φδ 1 a (2δ 1 b δ i j -δ i b δ 1 j -δ i 1 δ b j ) + φ(2δ ab δ i j -δ i b δ a j -δ i a δ b j ). If a = b = 1, D 2 11 J i j = -2δ i j + r 2 0 φ(δ i j -δ i 1 δ 1 j ) + 4r 0 φ(δ i j -δ i 1 δ 1 j ) + 2φ(δ i j -δ i 1 δ 1 j ) = -2δ i 1 δ 1 j + (r 2 0 φ + 4r 0 φ + 2φ -2)(δ i j -δ i 1 δ 1 j ) = -2δ i 1 δ 1 j + (-2r 0 tan r 0 sec 2 r 0 -2 sec 2 r 0 )(δ i j -δ i 1 δ 1 j ).
Notice that D 2 11 J i j = 0 if i ̸ = j. Therefore, it suffices to consider the case i = j.

D 2 11 J i i = -2δ i 1 -2(1 -δ i 1 ) + B(δ) = -2 + B(δ).
Together with (5.15), we deduce the first inequality in (5.8).

If a = 1, b > 1, or a > 1, b = 1. In view of D 2 ab J i j = D 2 ba J i j , we only need to consider the case a = 1, b > 1. D 2 1b J i j = -(r 0 φ + φ)(δ i b δ 1 j + δ i 1 δ b j ).
As a consequence, D 2 1b J i j ̸ = 0 except when i = 1, j = b or i = b, j = 1. By the symmetry of J , it suffices to consider the case i = b, j = 1.

D 2 1b J b 1 = -(r 0 φ + φ) = sec 2 r 0 -1 = B(δ).
Combining with (5.15), we get the second inequality in (5.8)

. If a, b > 1, D 2 ab J i j = 2(φ -1)δ ab δ i j + r 0 φδ ab (δ i j -δ i 1 δ 1 j ) - φ(δ i b δ a j + δ i a δ b j ).
Observe that D 2 ab

J i j ̸ = 0 except when a = b, i = j or a ̸ = b, i = a, j = b or a ̸ = b, i = b, j = a.
The case a = b, i = j :

D 2 aa J i i = 2(φ -1) + r 0 φ(1 -δ i 1 ) -2φδ i a = -2 tan r 0 r 0 + ( tan r 0 r 0 -sec 2 r 0 )(1 -δ i 1 ) + 2( tan r 0 r 0 -1)δ i a = -(1 -δ i 1 ) -δ i a + B(δ).
Combining with (5.15), we get (5.9). The case

a ̸ = b, i ̸ = j, i = a, j = b or a ̸ = b, i ̸ = j, i = b, j = a : D 2 ab J a b = D 2 ab J b a = -φ = tan r 0 r 0 -1 = -1 + B(δ).
Together with (5.15), we derive (5.10). This ends the proof of Proposition 5.1.

Proof of Theorem 5.1

In this subsection, we are going to prove Theorem 5.1. Assume that the curvatures of M satisfy (1.3) and (1.4). Fix any couple (k, α) ∈ N × (0, 1), with k ≥ 2. Let (ρ 0 dvol, ρ 1 dvol) be a couple of C k,α positive Borel probability measures on M.

As mentioned in Chapter 3, the MTW tensor on nearly spherical manifold satisfies the A3S condition. Thus the condition of Lemma 5.3 is satisfied. Granted Lemma 5.1, to complete Theorem 5.1, it is sufficient to prove (5.2).

Fix (t, m) ∈ I × M. Note that the left side in (5.2) is related to the initial Jacobi matrix J 0 , i.e. det d ∇mut exp m = det J 0 (m, ∇ m u t , 1).

It is also useful to recall that the gradient of u t at m locates in the injectivity domain at m. 4 . Thus, without loss of generality, we assume that there exists at least a point such that the length of gradient ∇u t at that point is not less than 3π 4 . Proof of Theorem 5.1. We assume that there exists at least a point such that the length of gradient ∇u t at that point is not less than 3π 4 . We will use the method of maximum principle to prove (5.2). We need to constructed an appropriate test function.

Let J (m, ∇ m u t ) = -|∇ m u t | 2 S -1 (m, ∇ m u t , 1). Consider the minimization problem:

min{⟨J ξ, ξ⟩ : (m, ξ) ∈ T M, 3π 4 ≤ |∇u t | m , |ξ| m = 1, ξ⊥∇ m u t }.
Suppose that the minimum is attained at the point (m 0 , ξ 0 ). We consider the test function:

h (m, ξ) = ⟨J ξ, ξ⟩ + ⟨ξ, ∇u t ⟩ 2 |ξ| 2 -⟨ξ,∇ut⟩ 2 |∇ut| 2 .
Then h attains the minimum at the point (m 0 , ξ 0 ) in a neighborhood of the point (m 0 , ξ 0 ) in T M. To see this, let ξ ⊥ be the orthonormal part of ξ. Then

⟨J ξ ⊥ , ξ ⊥ ⟩ |ξ ⊥ | 2 = h (m, ξ) .
By continuity, the test function h attains the local minimum at the point (m 0 , ξ 0 ) in a neighborhood of the point (m 0 , ξ 0 ) in T M.

The minimum h(m 0 , ξ 0 ) has a nice explanation, that is the second eigenvalue of the self-adjoint operator J . Specifically, as h is bilinear on the orthogonal complement subspaces (∇ m u t ) ⊥ with respect to ξ, thus the minimum h(m 0 , ξ 0 ) is the second eigenvalue of the self-adjoint operator J (m 0 , ∇ m0 u t ) with the associated eigenvector ξ 0 .

As a consequence of the above explanation, a necessary condition for (5.2) is that the minimum h(m 0 , ξ 0 ) has a positive lower bound. Thus (5.2) is transformed into the positive lower bound of h(m 0 , ξ 0 ). Notice that the minimum h(m 0 , ξ 0 ) has to be positive. To see this, from the Hessian Comparison Theorem, we know that -S ⊥ is not less than -r0 cos r0 sin r0 I n-1 which is positive definite when r 0 = |∇ m0 u t | ∈ ( π 2 , π). Thus the minimum h(m 0 , ξ 0 ) is positive.

In view of (5.4), we deduce r 0 sin r 0 cos r 0 -Cε ≤ h(m 0 , ξ 0 ) ≤ -r 0 sin r 0 cos r 0 + Cε, (5.17) where r 0 ≥ 3π 4 . Since the real value function -r sin r cos r is decreasing in ( π 2 , π), thus the right inequality infers that the minimum h(m 0 , ξ 0 ) has a positive upper bound. If h(m 0 , ξ 0 ) has a positive lower bound which is independent of the densities, by choosing ε sufficiently small, the right inequality also infers that r 0 ≤ π -δ for some δ > 0. This is the uniform gradient estimate.

In order to differentiate the test function h, it needs to rule out the boundary case. Since the function -r sin r cos r is decreasing in ( π 2 , π), the left inequality in (5.17) ensures that we can assume that r 0 > πδ, 0 < δ < π 4 . Henceforth, we will drop freely the subscript t. Take the Fermi coordinate system x along the geodesic exp m0 (s

∇m 0 u |∇m 0 u| ).

Some local notations

Components of some tensors in x will be denoted by:

grad u = ∇ i u(m) ∂ ∂x i , ∇ 2 m u = ∇ i j u(m)dx j ⊗ ∂ ∂x i , S = S i j (m, ν, 1)dx j ⊗ ∂ ∂x i , J = J i j (m, ν)dx j ⊗ ∂ ∂x i , H = H i j (m)dx j ⊗ ∂ ∂x i , F = F i j (m)dx j ⊗ ∂ ∂x i , where, H i j = ∇ i j u + S i j , H i k F k j = δ i j .
Fix ξ ∈ T m M. The coordinate of ξ in the Fermi coordinate system x is denoted by ξ = ξ i ∂ i . Then ⟨J ξ, ξ⟩ = J a b g ap ξ b ξ p , |ξ| 2 = g ab ξ a ξ b , ⟨ξ, ∇u⟩ = ξ a ∇ a u.

(5.18)

In the following all terms are evaluated at the point (x, v) = (0, r 0 , 0). It will be implicitly understood throughout the calculations. The components of ξ 0 are denoted by ξ i 0 , i.e. ξ 0 = ξ i 0 ∂ i , ξ 1 0 = 0. The first derivative condition Differentiating the test function h with respect to x i , the first derivative condition for the critical point could be read as:

(∂ i J α β + D k J α β ∇ k i u)ξ α 0 ξ β 0 = 0. (5.19)
The second derivative condition Differentiating twice on the test function h with respect to x i and x j respectively, the second derivative condition read as follows:

0 ≤ I 1 + II 1 + III 1 + IV 1 + V 1 , (5.20)
where

I 1 = -⟨J ξ 0 , ξ 0 ⟩F i j ∂ 2 ij g αβ ξ α 0 ξ β 0 + F i j ∂ 2 ij g αk J k β ξ α 0 ξ β 0 , II 1 = -F i j ∂ j Γ k il ∇ l uD k J α β ξ α 0 ξ β 0 + F i j ∂ 2 ij J α β ξ α 0 ξ β 0 , III 1 = 2F i j ∇ j k u∂ i D k J α β ξ α 0 ξ β 0 , IV 1 = 2(1 + 1 r 2 0 ⟨J ξ 0 , ξ 0 ⟩)F i j ∇ α i u∇ j β uξ α 0 ξ β 0 + F i j ∇ k i u∇ j l uD 2 kl J α β ξ α 0 ξ β 0 , V 1 = F i j ∂ j ∇ k i uD k J α β ξ α 0 ξ β 0 .
At the point m 0 , the potential function u satisfies the equation:

det J 0 det(H i j ) = ρ 0 ρ t . (5.21)
It is clear that the positive definiteness of the the matrix (H i j ) implies that ∇ 1 1 u is strictly greater than -1. We will also require the following expression:

S(0, r 0 , 0, 1) = [ 1 0 0 S α β ] . ( 5.22)
Note that -S ⊥ = -(S α β ) has a uniform lower bound under the curvature assumption (1.3). Indeed, from the Hessian Comparison Theorem, we know that -S ⊥ is not less than -r0 cos r0 sin r0 I n-1 . Making use of the fact that the real function -t cos t sin t is increasing in ( π 2 , π), we have for r 0 ≥ 3π 4 ,

-S ⊥ ≥ 3π 4 I n-1 .
(5.23)

We will calculate each term from I 1 to V 1 . The term I 1 Since g ij = δ ij on the axis and J 1 α = 0 at the point (0, r 0 , 0), the term I 1 can be written as:

I 1 = -⟨J ξ 0 , ξ 0 ⟩F φ ψ ∂ 2 φψ g αβ ξ α 0 ξ β 0 + F φ ψ ∂ 2 φψ g αι J ι β ξ α 0 ξ β 0 .
Making use of (1.14), it follows that

I 1 = 2 3 ⟨J ξ 0 , ξ 0 ⟩F φ ψ R φαψβ ξ α 0 ξ β 0 - 2 3 F φ ψ R φαψι J ι β ξ α 0 ξ β 0 .
Together with the curvature assumption (1.4), the positive definiteness of (F i j ) and the uniform bound for the norm of J , we have

I 1 ≤ 2 3 ⟨J ξ 0 , ξ 0 ⟩F α α - 2 3 ⟨J ξ 0 , ξ 0 ⟩F α β ξ α 0 ξ β 0 - 2 3 ⟨J ξ 0 , ξ 0 ⟩F α α + 2 3 F α φ J φ β ξ α 0 ξ β 0 + CεF α α = - 2 3 ⟨J ξ 0 , ξ 0 ⟩F α β ξ α 0 ξ β 0 + 2 3 F α φ J φ β ξ α 0 ξ β 0 + CεF α α ≤ - 2 3 ⟨(J -J )ξ 0 , ξ 0 ⟩F α β ξ α 0 ξ β 0 + CεF α α ,
where the last inequality follows from (5.4). By virtue of (5.4) and the positive definiteness of (F i j ) again, we derive that there exists a universal constant such that the following upper bound holds:

I 1 ≤ CεF α α .
(5.24)

The term II 1 There are two terms in II 1 . Proposition 5.1 derives that the second term is bounded by

π 2 F α α + C(ε + δ)F i i .
We mainly deal with the first term. By the first expression in (1.15) and ∇u = (r 0 , 0) at the point m 0 , we get

-F i j ∂ j Γ k il ∇ l uD k J α β ξ α 0 ξ β 0 = -r 0 F i j R k ij1 D k J α β ξ α 0 ξ β 0 = -r 0 F φ ψ R 1 φψ1 D 1 J α β ξ α 0 ξ β 0 -r 0 F i j R φ ij1 D φ J α β ξ α 0 ξ β 0 ≤ -πr 0 F α α + C(ε + δ)F i i ,
where the last inequality holds because of (1.4) and Proposition 5.1.

Thus we derive that there exists a positive constant C > 0 such that the following estimate holds:

II 1 ≤ π(π -r 0 )F α α + C(ε + δ)F i i ≤ C(ε + δ)F i i .
(5.25)

The term III 1 Using that F i j ∇ j k u = δ i k -F i j S j k , the term III 1 becomes:

III 1 = 2∂ k D k J α β ξ α 0 ξ β 0 -2F i j S j k ∂ i D k J α β ξ α 0 ξ β 0 .
It is easy to see that the first term is bounded by Cε by Proposition 5.1. For the second term, S j k is unbounded as the gradient goes to the conjugate locus. From FS = S -1 SFS and the boundedness of S -1 , we infer the existence of a positive constant C such that:

III 1 ≤ Cε(1 + F i j S k i S j k ).
Let us observe that the following identity holds:

F i j ∇ k i u∇ j l u = ∇ k l u -S k l + F i j S k i S j l .
(5.26)

In particular,

F i j ∇ α i u∇ j β u = H α β -2S α β + F i j S α i S j β
. By (5.23), (5.22) and the positive definiteness of (H i j ), (-S α β ), we get

III 1 ≤ Cε(F 1 1 + F i j ∇ α i u∇ j α u).
(5.27)

The term IV 1 Splitting the negative term F i j ∇ k i u∇ j l uD 2 kl J α β ξ α 0 ξ β 0 into four parts, we get

IV 1 = F i j ∇ 1 i u∇ j 1 uD 2 11 J α β ξ α 0 ξ β 0 + 2F i j ∇ 1 i u∇ j ι uD 2 1ι J α β ξ α 0 ξ β 0 + F i j ∇ φ i u∇ j φ uD 2 φφ J α β ξ α 0 ξ β 0 + ∑ φ̸ =ψ F i j ∇ φ i u∇ j ψ uD 2 φψ J α β ξ α 0 ξ β 0 + 2(1 + 1 r 2 0 ⟨J ξ 0 , ξ 0 ⟩)F i j ∇ α i u∇ j β uξ α 0 ξ β 0 .
Using Proposition 5.1, we infer that there exists a positive constant C such that

IV 1 ≤ [C(ε + δ) -2]F i j ∇ 1 i u∇ j 1 u + [C(ε + δ) + 2 r 2 0 ⟨J ξ 0 , ξ 0 ⟩ -1]F i j ∇ α i u∇ j α u.
In view of (5.26), it follows that

F i j ∇ 1 i u∇ j 1 u = H 1 1 -2 + F 1 1 .
For ε and δ small enough(ε, δ < 1 C ), by the positive definiteness of (H i j ), we get

IV 1 ≤ [C(ε + δ) -2](F 1 1 -2) + (5.28) [C(ε + δ) + 2 r 2 0 ⟨J ξ 0 , ξ 0 ⟩ -1]F i j ∇ α i u∇ j α u.
The term V 1 The term V 1 involves the third derivatives of u. After commuting the third derivatives of u, the term V 1 can be written:

V 1 = F i j (∂ k ∇ j i u + r 0 R k i1j )D k J α β ξ α 0 ξ β 0 .
We first compute the third derivative. In the Fermi coordinate system x, the determinant of the positive matrix (g ij ) 1≤i,j≤m is denoted by |g|. Recall the equation ( 3), by definition of det d ∇mu exp m , the potential function u satisfies the Monge-Ampère type equation det Hess (c)

u = √ |g|(x)ρ 0 (x) √ |g|(X) det(D v X)ρ t (x) . ( 5.29)
By taking the logarithm and differentiating the associated equation with respect to the variable x k at the point (0, r 0 , 0), we obtain

F i j ∂ k ∇ j i u = ∂ k ρ 0 ρ 0 - (1 -t)∂ k ρ 0 + t∂ p ρ 1 (∂ k X p + D l X p ∇ l k u) ρ t - B p q (∂ k D p X q + D 2 pl X q ∇ l k u) + F i j S q i S j p ∂ k A p q + F i j S q i S j p D l A p q ∇ l k u = ∂ k ρ 0 ρ 0 - (1 -t)∂ k ρ 0 + t∂ p ρ 1 ∂ k X p ρ t -B p q ∂ k D p X q + F i j S q i S j p ∂ k A p q + (- t∂ p ρ 1 D l X p ρ t -B p q D 2 pl X q + F i j S q i S j p D l A p q )∇ l k u,
where the matrix (B p q ) is the inverse of the matrix (D j X i ) and (A p q ) is the inverse of the matrix (S i j ). Thus we get the following simplified expressions:

V 1 = [ ∂ k ρ 0 ρ 0 - (1 -t)∂ k ρ 0 + t∂ p ρ 1 ∂ k X p ρ t -B p q ∂ k D p X q + F i j S q i S j p ∂ k A p q ]D k J α β ξ α 0 ξ β 0 + (- t∂ p ρ 1 D l X p ρ t -B p q D 2 pl X q + F i j S q i S j p D l A p q )∇ l k uD k J α β ξ α 0 ξ β 0 + r 0 F i j R k i1j D k J α β ξ α 0 ξ β 0 .
By the critical condition (5.19), Lemma 4.3, Lemma 4.4, Proposition 5.1, Cauchy-Schwarz inequality, we deduce that there exists a positive constant C such that

V 1 ≤ CM + C(ε + δ)F 1 1 + [C(ε + δ) -π 2 ]F α α + (5.30) C(ε + δ)F i j ∇ α i u∇ j α u, where M = max{max M |d log ρ 0 |, max M |d log ρ 1 |}.
Plugging the upper bounds (5.24),(5.25),(5.27),(5.28),(5.30) into the inequality (5.20), we obtain the following inequality:

0 ≤ 4 + CM + [4C(ε + δ) -2]F 1 1 + [3C(ε + δ) -π 2 ]F α α + [3C(ε + δ) + 2 r 2 0 ⟨J ξ 0 , ξ 0 ⟩ -1]F i j ∇ α i u∇ j α u = 4 + CM + [4C(ε + δ) -2]F 1 1 + [3C(ε + δ) -π 2 ]F α α + [3C(ε + δ) - 1 2 ]F i j ∇ α i u∇ j α u + ( 2 r 2 0 ⟨J ξ 0 , ξ 0 ⟩ - 1 2 )F i j ∇ α i u∇ j α u. Fix ε < 1 24C , δ < 1 24C
. Therefore, we get the inequality:

0 ≤ 4 + CM - 5 3 F 1 1 + ( 1 4 -π 2 )F α α - 1 4 F i j S α i S j α + ( 2 r 2 0 ⟨J ξ 0 , ξ 0 ⟩ - 1 2 )F i j ∇ α i u∇ j α u ≤ 4 + CM - 1 4 F i j S k i S j k + ( 2 r 2 0 ⟨J ξ 0 , ξ 0 ⟩ - 1 2 )F i j ∇ α i u∇ j α u. If ⟨J ξ 0 , ξ 0 ⟩ ≥ (π-δ) 2 4
, there is nothing to prove. We only have to consider the case

⟨J ξ 0 , ξ 0 ⟩ ≤ (π-δ) 2 4
. Observe that the last term is non-positive in this case, thus

0 ≤ 4 + CM - 1 4 F i j S k i S j k .
By the inequality of arithmetic and geometric means and the equation (5.21), we get that there exists a positive number C such that

F i j S k i S j k ≥ C(min{1, ρ 1 ρ 0 }) 1 n 1 ⟨J ξ 0 , ξ 0 ⟩ 1 n . Thus ⟨J ξ 0 , ξ 0 ⟩ is bounded below by δ 0 = Cn min{1, ρ 1 ρ 0 } (16+4CM) n .
In conclusion, we just need to choose ε, δ such that 0 < ε < 1 24C , 0 < δ < min{ 1 24C , π -2 √ δ 0 }, then the estimate (5.2) is proved. By the continuity method, we prove Theorem 5.1.

The smoothness of the optimal transport map on product manifolds

In this section, we investigate the smoothness of the optimal transport map on Riemannian product manifold of nearly spherical manifolds. By Lemma 5.1, the regularity of the optimal transport map reduces to (5.2) and (5.3).

Let (M 1 , g) and (M 2 , ĝ) be two closed Riemannian manifolds of dimension n 1 ≥ 2 and n 2 ≥ 2 respectively. Suppose that both M 1 and M 2 satisfy (1.3) and (1.4).

It is known that the sectional curvatures of M 1 × M 2 are non-negative and may vanish. Using Corollary 3.1, the corresponding MTW tensor satisfies A3W condition and may vanish on some directions. Moreover, the MTW tensor is non-negative.

The non-trivial cut locus and the vanishing of MTW tensor are the main obstacles of the smoothness. The strategy is to establish that the optimal transport map uniformly stay away from the cut locus by the maximum principle. This result allows to derive the uniformly second order derivative estimate. Then the method of continuity implies the smoothness of the optimal transport map.

Uniform stay away estimate

In this subsection, we will settle the estimate (5.2). Even through the cut locus of M 1 × M 2 is non-trivial, we will prove that the optimal map uniformly stay away from the cut locus by the method of maximum principle.

Fix any couple (k, α) ∈ N × (0, 1), with k ≥ 2. Let (ρ 0 dvol, ρ 1 dvol) be C k,α positive Borel probability measures on

M 1 × M 2 . Fix t ∈ I, m = ( m, m) ∈ M 1 × M 2 .
Note that the left side in (5.2) is related to the initial matrix J 0 . Actually,

det d ∇mut exp m = det J 0 (m, ∇ m u t , 1) = det J0 ( m, ∇ mu t , 1) det Ĵ0 ( m, ∇ mu t , 1),
where the second equality follows from Lemma 5.4.

It is also useful to mention that the gradient of u t at m locates in the injectivity domain at m. Then by the Bishop's theorem, det J0 ( m, ∇ mu t , 1) and det Ĵ0 ( m, ∇ mu t , 1) are all uniformly bounded from above by 1 if both M 1 and M 2 have non-negative Ricci curvatures.

By the description of Jacobi fields in Section 1.2.1, we know that the discriminants det J0 ( m, ∇ mu t , 1) and det Ĵ0 ( m, ∇ mu t , 1) are all positive. Thus the positive lower bound of det d ∇mut exp m is equivalent to the positive lower bounds of the discriminants det J0 ( m, ∇ mu t , 1) and det Ĵ0 ( m, ∇ mu t , 1). But det d ∇mut exp m may not has a positive lower bound in general. Since det J 0 (m, ν, 1) vanishes if (and only if) exp m ν is conjugate to m, so the estimate (5.2) is not obvious, for instance, on the Riemannian products of the round sphere

S n1 × S n2 , det d ∇mut exp m = ( sin | ∇ m ut| | ∇ m ut| ) n1-1 ( sin | ∇ m ut| | ∇ mut | ) n2-1 is close to zero as | ∇ mu t | approaches π or | ∇ mu t | approaches π.
We will use the method of maximum principle to prove (5.2). It needs to construct an appropriate test function.

Henceforth, we will drop freely the subscript t.

Before showing the estimate (5.2), we give a Claim first.

( ξ, ξ) ∈ T mM 1 × T mM 2 , | ξ| m = 1 = | ξ| m, g( ξ, ∇ mu) = 0 = ĝ( ξ, ∇ mu), | ∇ mu| ≥ 3π 4 , | ∇ mu| ≥ 3π 4 }.
has a positive lower bound δ 1 which depends on the densities and n 1 , n 2 .

Notice that both ⟨J ξ, ξ⟩ and ⟨J ξ, ξ⟩ have to be positive. To see this, from the Hessian Comparison Theorem, it follows that -S ⊥ (the restriction of -S to (R ∇ mu) ⊥ in T mM 1 ) is not less than -r1 cos r1 sin r1 I n1-1 which is positive definite when r 1 = | ∇ mu| ∈ ( π 2 , π). By definition of J , we know that ⟨J ξ, ξ⟩ is positive. Similarly, ⟨J ξ, ξ⟩ is also positive.

By virtue of 1) in Theorem 4.1, there exists a positive number C such that, for any .33) where

( ξ, ξ) ∈ T mM 1 × T mM 2 , | ξ| m = 1 = | ξ| m, - r 1 sin r 1 cos r 1 -Cε ≤ ⟨J ξ, ξ⟩ ≤ - r 1 sin r 1 cos r 1 + Cε, (5.32) - s 1 sin s 1 cos s 1 -Cε ≤ ⟨J ξ, ξ⟩ ≤ - s 1 sin s 1 cos s 1 + Cε. ( 5
r 1 = | ∇ mu t | ≥ 3π 4 and s 1 = | ∇ mu t | ≥ 3π 4 .
Since the function -r sin r cos r is decreasing in ( π 2 , π), thus the right inequalities in (5.32) and (5.33) infer that both ⟨J ξ, ξ⟩ and ⟨J ξ, ξ⟩ are bounded from above by some positive constant. In such case, we infer that max{r 1 , s 1 } ≤ π -δ for some δ > 0. These are the uniform gradient estimates. Proof of Claim 5.1. It is clear that the minimum of (⟨J ξ, ξ⟩ + π 2 -| ∇ mu| 2 )(⟨J ξ, ξ⟩ + π 2 -| ∇ mu| 2 ) is attained and finite. Let ( m0 , m0 , ξ0 , ξ0 ) be the minimum point. We consider the test function:

h (m, ξ) = log( ⟨J ξ, ξ⟩ + ⟨ ξ, ∇u⟩ 2 | ξ| 2 - ⟨ ξ, ∇u⟩ 2 | ∇u| 2 + π 2 -| ∇u| 2 ) + log( ⟨J ξ, ξ⟩ + ⟨ ξ, ∇u⟩ 2 | ξ| 2 - ⟨ ξ, ∇u⟩ 2 | ∇u| 2 + π 2 -| ∇u| 2 ).
Then h attains the minimum at the point ( m0 , m0 , ξ0 , ξ0 ) in a neighborhood of the point ( m0 , m0 , ξ0 , ξ0 ) in T (M 1 × M 2 ). To see this, let ξ⊥ be the orthonormal part of ξ in T m0 M 1 and ξ⊥ be the orthonormal part of ξ in

T m0 M 2 . Then log( ⟨J ξ⊥ , ξ⊥ ⟩ | ξ⊥ | 2 + π 2 -| ∇u| 2 ) + log( ⟨J ξ⊥ , ξ⊥ ⟩ | ξ⊥ | 2 + π 2 -| ∇u| 2 ) = h (m, ξ) .
By continuity, we obtain that the test function h attains the local minimum at the point ( m0 , m0 , ξ0 , ξ0 ) in a neighborhood of the point ( m0 , m0 , ξ0 , ξ0 ) in T (M 1 × M 2 ). Note that the terms ⟨J ξ0 , ξ0 ⟩ and ⟨J ξ0 , ξ0 ⟩ are all eigenvalues of the self adjoint operator J .

Set for short:

r 1 = | ∇ m0 u|, s 1 = | ∇ m0 u|, r 2 = ⟨J ξ0 , ξ0 ⟩ > 0, s 2 = ⟨J ξ0 , ξ0 ⟩ > 0, r 3 = π 2 -r 2 1 , s 3 = π 2 -s 2 1 .
Let x be the Fermi coordinate system in M 1 along the geodesic exp m0 (s ∇ m0 u r1 ) and y be the Fermi coordinate system in M 2 along the geodesic exp m0 (s ∇ m0 u s1 ). Then z = (x, y) is the coordinate system in M 1 × M 2 . The associated coordinate system in tangent bundle T (M 1 × M 2 ) is denoted by (z, v).

In the following all terms are evaluated at the point (z, v) = (0, r 1 , 0, s 1 , 0). It will be implicitly understood throughout the calculations. The components of ξ 0 = ( ξ0 , ξ0 ) are denoted by ( ξi1

0 , ξi2 0 ) i.e. ξ0 = n1 ∑ i1=1 ξi1 0 ∂ ∂x i1 , ξ0 = n1+n2 ∑ i2=n1+1 ξi2 0 ∂ ∂y i2 .
(5.34)

It is clear that ξ1 = ξ(n1+1) = 0. We are in position to calculate the derivatives of the test function h. It is clear that the Claim 5.1 is proved if max{r 1 , s 1 } ≤ η 0 for η 0 ∈ (0, π). Without loss of generality, we shall assume that min{r 1 , s 1 } > πδ where δ ∈ (0, π 4 ) is determined later. We give the following notations:the Latin indices run over {1,

• • • , n 1 + n 2 }, the indices i 1 , j 1 , • • • run over {1, • • • , n 1 }, the indices i 2 , j 2 , • • • run over {n 1 + 1, • • • , n 1 + n 2 }, the indices α 1 , β 1 , • • • run over {2, • • • , n 1 }, and the indices α 2 , β 2 , • • • run over {n 1 + 2, • • • , n 1 + n 2 }.

The first derivative condition

By differentiating the test function h with respect to z i , the first derivative condition could be read as:

1 r 2 + r 3 [ (∂ i J α1 β1 + D k J α1 β1 ∇ k i u) ξα1 0 ξβ1 0 -2s 1 ∇ n1+1 i u ] + 1 s 2 + s 3 [ (∂ i J α2 β2 + D k J α2 β2 ∇ k i u) ξα2 0 ξβ2 0 -2r 1 ∇ 1 i u ] = 0. ( 5

.35)

The second derivative condition Differentiating twice the test function h with respect to z i and z j respectively, the second derivative condition read as follows:

0 ≤ I 2 + II 2 + III 2 + IV 2 + V 2 + V I 2 , ( 5.36)
where

I 2 = - 1 (r 2 + r 3 ) 2 F i j [(∂ i J α1 β1 + D k J α1 β1 ∇ k i u) ξα1 0 ξβ1 0 - 2s 1 ∇ n1+1 i u][(∂ j J ϕ1 ψ1 + D l J ϕ1 ψ1 ∇ l j u) ξϕ1 0 ξψ1 0 -2s 1 ∇ n1+1 j u] - 1 (s 2 + s 3 ) 2 F i j [(∂ i J α2 β2 + D k J α2 β2 ∇ k i u) ξα2 0 ξβ2 0 - 2r 1 ∇ 1 i u][(∂ j J ϕ2 ψ2 + D l J ϕ2 ψ2 ∇ l j u) ξϕ2 0 ξψ2 0 -2r 1 ∇ 1 j u], II 2 = 1 r 2 + r 3 F i j [∂ 2 ij g α1k J k β1 ξα1 0 ξβ1 0 -r 2 ∂ 2 ij g α1β1 ξα1 0 ξβ1 0 + (2∂ j Γ p2+η iq2 -∂ 2 ij g p2q2 )∇ p2 u∇ q2 u] + 1 s 2 + s 3 F i j [∂ 2 ij g α2k J k β2 ξα2 0 ξβ2 0 -s 2 ∂ 2 ij g α2β2 ξα2 0 ξβ2 0 + (2∂ j Γ p1 iq1 -∂ 2 ij g p1q1 )∇ p1 u∇ q1 u], III 2 = 1 r 2 + r 3 F i j (∂ 2 ij J α1 β1 ξα1 0 ξβ1 0 -∂ j Γ k il ∇ l uD k J α1 β1 ξα1 0 ξβ1 0 ) + 1 s 2 + s 3 F i j (∂ 2 ij J α2 β2 ξα2 0 ξβ2 0 -∂ j Γ k li ∇ l uD k J α2 β2 ξα2 0 ξβ2 0 ), IV 2 = 2 r 2 + r 3 F i j ∇ j k u∂ i D k J α1 β1 ξα1 0 ξβ1 0 + 2 s 2 + s 3 F i j ∇ j k u∂ i D k J α2 β2 ξα2 0 ξβ2 0 , V 2 = 1 r 2 + r 3 F i j [∇ k i u∇ j l uD 2 kl J α1 β1 ξα1 0 ξβ1 0 + 2(1 + r 2 r 2 1 )∇ α1 i u∇ j β1 u ξα1 0 ξβ1 0 -2∇ k2 i u∇ j k2 u] + 1 s 2 + s 3 F i j [∇ k i u∇ j l uD 2 kl J α2 β2 ξα2 0 ξβ2 0 + 2(1 + s 2 s 2 1 )∇ α2 i u∇ j β2 u ξα2 0 ξβ2 0 -2∇ k1 i u∇ j k1 u], V I 2 = 1 r 2 + r 3 F i j (∂ j ∇ k i uD k J α1 β1 ξα1 0 ξβ1 0 -2∂ j ∇ k2 i u∇ k2 u) + 1 s 2 + s 3 F i j (∂ j ∇ k i uD k J α2 β2 ξα2 0 ξβ2 0 -2∂ j ∇ k1 i u∇ k1 u).
The potential function u at the point m 0 satisfies the equation: .37) Notice that the positive definiteness of the matrix (H i j ) implies that both ∇ 1 1 u and ∇ (n1+1) (n1+1) u are all greater than -1. We will also require the expression: The term II 2 Since g ij = δ ij on the axis and J 1 α1 = J (n1+1) α2

det J 0 det(H i j ) = ρ 0 ρ t . ( 5
S(0, r 1 , 0, s 1 , 0, 1) =     1 0 0 0 0 S α1 β1 0 0 0 0 1 0 0 0 0 S α2 β2     . ( 5
= J j2 i1 = J i1 j2 = 0 at the point (0, r 1 , 0, s 1 , 0), the term II 1 can be written as:

II 2 = 1 r 2 + r 3 [ n1 ∑ i,j,k=2 F i j ∂ 2 ij g α1k J k β1 ξα1 0 ξβ1 0 -r 2 n1 ∑ i,j=2 F i j ∂ 2 ij g α1β1 ξα1 0 ξβ1 0 + s 2 1 n1+n2+η ∑ i,j=n1+2 F i j (2∂ j Γ (n1+1) i(n1+1) -∂ 2 ij g (n1+1)(n1+1) )] + 1 s 2 + s 3 [ n1+n2 ∑ i,j,k=n1+2 F i j ∂ 2 ij g α2k J k β2 ξα2 0 ξβ2 0 -s 2 n1+n2 ∑ i,j=n1+2 F i j ∂ 2 ij g α2β2 ξα2 0 ξβ2 0 + r 2 1 n1 ∑ i,j=2 F i j (2∂ j Γ 1 i1 -∂ 2 ij g 11 )].
By (1.14), it follows that

II 2 = 1 r 2 + r 3 [- 2 3 n1 ∑ i,j,k=2 F i j R iα1jk J k β1 ξα1 0 ξβ1 0 + 2 3 r 2 n1 ∑ i,j=2 F i j R iα1jβ1 ξα1 0 ξβ1 0 + s 2 1 n1+n2 ∑ i,j=n1+2 F i j (2R (n1+1) ij(n1+1) + 2R (n1+1)i(n1+1)j )] + 1 s 2 + s 3 [- 2 3 n1+n2 ∑ i,j,k=n1+2 F i j R iα2jk J k β2 ξα2 0 ξβ2 0 + 2 3 s 2 n1+n2 ∑ i,j=n1+2 F i j R iα2jβ2 ξα2 0 ξβ2 0 + r 2 1 n1 ∑ i,j=2 F i j (2R 1 ij1 + 2R 1i1j )] = 1 r 2 + r 3 (- 2 3 n1 ∑ i,j,k=2 F i j R iα1jk J k β1 ξα1 0 ξβ1 0 + 2 3 r 2 n1 ∑ i,j=2 F i j R iα1jβ1 ξα1 0 ξβ1 0 ) + 1 s 2 + s 3 (- 2 3 n1+n2 ∑ i,j,k=n1+2 F i j R iα2jk J k β2 ξα2 0 ξβ2 0 + 2 3 s 2 n1+n2 ∑ i,j=n1+2 F i j R iα2jβ2 ξα2 0 ξβ2 0 ).
Together with the curvature assumption (1.4), the positive definiteness of (F i j ) and the uniform bound for the norm of J , we have

II 2 ≤ 1 r 2 + r 3 (- 2 3 r 2 F α1 α1 + 2 3 F α1 φ1 J φ1 β1 ξα1 0 ξβ1 0 + 2 3 r 2 F α1 α1 - 2 3 r 2 F α1 β1 ξα1 0 ξβ1 0 + CεF α1 α1 ) + 1 s 2 + s 3 (- 2 3 s 2 F α2 α2 + 2 3 F α2 φ2 J φ2 β2 ξα2 0 ξβ2 0 + 2 3 s 2 F α2 α2 - 2 3 s 2 F α2 β2 ξα2 0 ξβ2 0 + CεF α2 α2 ) = 1 r 2 + r 3 ( 2 3 F α1 φ1 J φ1 β1 ξα1 0 ξβ1 0 - 2 3 r 2 F α1 β1 ξα1 0 ξβ1 0 + CεF α1 α1 ) + 1 s 2 + s 3 ( 2 3 F α2 φ2 J φ2 β2 ξα2 0 ξβ2 0 - 2 3 s 2 F α2 β2 ξα2 0 ξβ2 0 + CεF α2 α2 ) ≤ 1 r 2 + r 3 ( 2 3 ⟨( J -J ) ξ0 , ξ0 ⟩F α1 β1 ξα1 0 ξβ1 0 + CεF α1 α1 ) + 1 s 2 + s 3 ( 2 3 ⟨(J -J ) ξ0 , ξ0 ⟩F α2 β2 ξα2 0 ξβ2 0 + CεF α2 α2 ).
where the last inequality follows from (5.4). By virtue of (5.4) and the positive definiteness of (F i j ) again, we derive that there exists a universal constant C > 0 such that there holds:

II 2 ≤ Cε r 2 + r 3 F α1 α1 + Cε s 2 + s 3 F α2 α2 . ( 5.41)
The term III 2 Noting ∇u = (r 1 , 0, s 1 , 0) at the point m 0 , and using the first expression in (1.15) and the definition of J , it follows that

III 2 = 1 r 2 + r 3 ( n1 ∑ i,j=2 F i j ∂ 2 ij J α1 β1 ξα1 0 ξβ1 0 - r 1 n1 ∑ i,j,k=2 F i j R k ij1 D k J α1 β1 ξα1 0 ξβ1 0 ) + 1 s 2 + s 3 ( n1+n2 ∑ i,j=n1+2 F i j ∂ 2 ij J α2 β2 ξα2 0 ξβ2 0 - s 1 n1+n2 ∑ i,j,k=n1+2 F i j R k ij(n1+1) D k J α2 β2 ξα2 0 ξβ2 0 ).
In light of Proposition 5.1 and the curvature assumption (1.4), we obtain

III 2 ≤ 1 r 2 + r 3 [π(π -r 1 )F α1 α1 + C(ε + δ)F i1 i1 ] + 1 s 2 + s 3 [π(π -s 1 )F α1 α1 + C(ε + δ)F i2 i2 ].
Thus we derive that there exists a positive constant C such that the following estimate holds:

III 2 ≤ C(ε + δ) r 2 + r 3 F i1 i1 + C(ε + δ) s 2 + s 3 F i2 i2 .
(5.42)

The term IV 2 Making use of

F i j ∇ j k u = δ i k -F i j S j k , the term IV 2 becomes: IV 2 = 2 r 2 + r 3 (∂ k D k J α1 β1 ξα1 0 ξβ1 0 -F i j S j k ∂ i D k J α1 β1 ξα1 0 ξβ1 0 ) + 2 s 2 + s 3 (∂ k D k J α2 β2 ξα2 0 ξβ2 0 -F i j S j k ∂ i D k J α2 β2 ξα2 0 ξβ2 0 ) = 2 r 2 + r 3 ( n1 ∑ k=1 ∂ k D k J α1 β1 ξα1 0 ξβ1 0 - n1 ∑ i,j,k=1 F i j S j k ∂ i D k J α1 β1 ξα1 0 ξβ1 0 ) + 2 s 2 + s 3 ( n1+n2 ∑ k=n1+1 ∂ k D k J α2 β2 ξα2 0 ξβ2 0 - n1+n2 ∑ i,j,k=n1+1 F i j S j k ∂ i D k J α2 β2 ξα2 0 ξβ2 0 ).
According to Proposition 5.1, the identity FS = S -1 SFS and the boundedness of S -1 , it follows that

IV 2 ≤ Cε r 2 + r 3 (1 + F i j S k1 i S j k1 ) + Cε s 2 + s 3 (1 + F i j S k2 i S j k2 ).
Let us observe that the following identities holds:

F i j ∇ k i u∇ j l u = ∇ k l u -S k l + F i j S k i S j l = H k l -2S k l + F i j S k i S j l . ( 5.43)
By (5.38),(5.39) and the positive definiteness of (H i j ), (-S α1 β1 ), (-S α2 β2 ), we derive the existence of a positive constant C such that:

IV 2 ≤ Cε r 2 + r 3 (F 1 1 + F i j ∇ α1 i u∇ j α1 u) + Cε s 2 + s 3 (F (n1+1) (n1+1) + F i j ∇ α2 i u∇ j α2 u).
(5.44)

The term V 2 By definition of J , we get

V 2 = 1 r 2 + r 3 [ n1 ∑ k,l=1 F i j ∇ k i u∇ j l uD 2 kl J α1 β1 ξα1 0 ξβ1 0 + 2(1 + r 2 r 2 1 )F i j ∇ α1 i u∇ j β1 u ξα1 0 ξβ1 0 -2F i j ∇ k2 i u∇ j k2 u] + 1 s 2 + s 3 [ n1+n2 ∑ k,l=n1+1 F i j ∇ k i u∇ j l uD 2 kl J α2 β2 ξα2 0 ξβ2 0 + 2(1 + s 2 s 2 1 )F i j ∇ α2 i u∇ j β2 u ξα2 0 ξβ2 0 -2∇ k1 i u∇ j k1 u] = 1 r 2 + r 3 [(F i j ∇ 1 i u∇ j 1 uD 2 11 J α1 β1 ξα1 0 ξβ1 0 + 2F i j ∇ 1 i u∇ j ι1 uD 2 1ι1 J α1 β1 ξα1 0 ξβ1 0 + F i j ∇ φ1 i u∇ j φ1 uD 2 φ1φ1 J α1 β1 ξα1 0 ξβ1 0 + ∑ φ1̸ =ψ1 F i j ∇ φ1 i u∇ j ψ1 uD 2 φ1ψ1 J α1 β1 ξα1 0 ξβ1 0 )+ 2(1 + r 2 r 2 1 )F i j ∇ α1 i u∇ j β1 u ξα1 0 ξβ1 0 -2F i j ∇ k2 i u∇ j k2 u] + 1 s 2 + s 3 [(F i j ∇ n1+1 i u∇ j n1+1 uD 2 (n1+1)(n1+1) J α2 β2 ξα2 0 ξβ2 0 + 2F i j ∇ n1+1 i u∇ j ι2 uD 2 (n1+1)ι2 J α2 β2 ξα2 0 ξβ2 0 + F i j ∇ φ2 i u∇ j φ2 uD 2 φ2φ2 J α 2 β2 ξα 2 0 ξβ2 0 + ∑ φ2̸ =ψ2 F i j ∇ φ2 i u∇ j ψ2 uD 2 φ2ψ2 J α2 β2 ξα2 0 ξβ2 0 + 2(1 + s 2 s 2 1 )F i j ∇ α2 i u∇ j β2 u ξα2 0 ξβ2 0 -2F i j ∇ k1 i u∇ j k1 u].
Making use of Proposition 5.1, we infer that there exists a positive constant C such that the following esimate holds

V 2 ≤ 1 r 2 + r 3 {[C(ε + δ) -2]F i j ∇ 1 i u∇ j 1 u + [C(ε + δ) + 2r 2 r 2 1 -1]F i j ∇ α1 i u∇ j α1 u -2F i j ∇ k2 i u∇ j k2 u} + 1 s 2 + s 3 {[C(ε + δ) -2]F i j ∇ n1+1 i u∇ j n1+1 u + [C(ε + δ) + 2s 2 s 2 1 -1]F i j ∇ α2 i u∇ j α2 u -2F i j ∇ k1 i u∇ j k1 u}.
In view of (5.26), it follows that

F i j ∇ 1 i u∇ j 1 u = H 1 1 -2 + F 1 1 , F i j ∇ n1+1 i u∇ j n1+1 u = H n1+1 n1+1 -2 + F n1+1 n1+1 .
Choose ε and δ small enough(ε, δ < 1 C ). By the positive definiteness of (H i j ), (-S α1 β1 ) and (-S α2 β2 ), we have

V 2 ≤ 1 r 2 + r 3 {[C(ε + δ) -2](F 1 1 -2) + [C(ε + δ) + 2r 2 r 2 1 -1]F i j ∇ α1 i u∇ j α1 u -2(F n1+1 n1+1 -2) -2F i j ∇ α2 i u∇ j α2 u} + 1 s 2 + s 3 {[C(ε + δ) -2](F n1+1 n1+1 -2) + [C(ε + δ) + 2s 2 s 2 1 -1]F i j ∇ α2 i u∇ j α2 u -2(F 1 1 -2) -2F i j ∇ α1 i u∇ j α1 u} ≤ 1 r 2 + r 3 {8 + [C(ε + δ) -2]F 1 1 + [C(ε + δ) + 2r 2 r 2 1 -1]F i j ∇ α1 i u∇ j α1 u -2F n1+1 n1+1 -2F i j ∇ α2 i u∇ j α2 u} + 1 s 2 + s 3 {8 + [C(ε + δ) -2]F n1+1 n1+1 + [C(ε + δ) + 2s 2 s 2 1 -1]F i j ∇ α2 i u∇ j α2 u -2F 1 1 -2F i j ∇ α1 i u∇ j α1 u}.
(5.45)

The term V I 2 The term V I 2 involves the third derivatives of u. After commuting the third derivatives of u, the term V I 2 can be read as:

V I 2 = 1 r 2 + r 3 [F i j (∂ k ∇ j i u + r 1 R k i1j )D k J α1 β1 ξα1 0 ξβ1 0 - 2s 1 F i j (∂ n1+1 ∇ j i u + s 1 R n1+1 i(n1+1)j )] + 1 s 2 + s 3 [F i j (∂ k ∇ j i u + s 1 R k i(n1+1)j )D k J α2 β2 ξα2 0 ξβ2 0 - 2r 1 F i j (∂ 1 ∇ j i u + r 1 R 1 i1j )
]. We compute the third derivative. In the Fermi coordinates, the determinant of the positive matrix (g ij ) 1≤i,j≤m is denoted by |g|. Recall the equation (5.29). By definition of det d ∇mu exp m , the potential function u satisfies the Monge-Ampère type equation det Hess (c) By taking the logarithm and differentiating the associated equation with respect to the partial variable z k at the point (0, r 1 , 0, s 1 , 0), we obtain

F i j ∂ k ∇ j i u = ∂ k ρ 0 ρ 0 - (1 -t)∂ k ρ 0 + t∂ p ρ 1 (∂ k Z p + D l Z p ∇ l k u) ρ t - B p q (∂ k D p Z q + D 2 pl Z q ∇ l k u) + F i j S q i S j p ∂ k A p q + F i j S q i S j p D l A p q ∇ l k u = ∂ k ρ 0 ρ 0 - (1 -t)∂ k ρ 0 + t∂ p ρ 1 ∂ k Z p ρ t -B p q ∂ k D p Z q + F i j S q i S j p ∂ k A p q + (- t∂ p ρ 1 D l Z p ρ t -B p q D 2 pl Z q +
F i j S q i S j p D l A p q )∇ l k u, where the matrix (B p q ) is the inverse of the matrix (D j Z i ) and (A p q ) is the inverse of the matrix (S i j ). Thus we get the following simplified expressions:

V I 2 = 1 r 2 + r 3 {[ ∂ k ρ 0 ρ 0 - (1 -t)∂ k ρ 0 + t∂ p ρ 1 ∂ k X p ρ t -B p q ∂ k D p X q + F i j S q i S j p ∂ k A p q ]D k J α1 β1 ξα1 0 ξβ1 0 + (- t∂ p ρ 1 D l X p ρ t -B p q D 2 pl X q + F i j S q i S j p D l A p q )∇ l k uD k J α1 β1 ξα1 0 ξβ1 0 + r 1 F i j R k i1j D k J α1 β1 ξα1 0 ξβ1 0 - 2s 1 [ ∂ n1+1 ρ 0 ρ 0 - (1 -t)∂ n1+1 ρ 0 + t∂ p ρ 1 ∂ n1+1 X p ρ t -B p q ∂ n1+1 D p X q + F i j S q i S j p ∂ n1+1 A p q ] + 2s 1 ( t∂ p ρ 1 D l X p ρ t + B p q D 2 pl X q - F i j S q i S j p D l A p q )∇ l n1+1 u -2s 2 1 F i j R n1+1 i(n1+1)j } + 1 s 2 + s 3 {[ ∂ k ρ 0 ρ 0 - (1 -t)∂ k ρ 0 + t∂ p ρ 1 ∂ k X p ρ t -B p q ∂ k D p X q + F i j S q i S j p ∂ k A p q ]D k J α2 β2 ξα2 0 ξβ2 0 + (- t∂ p ρ 1 D l X p ρ t -B p q D 2 pl X q + F i j S q i S j p D l A p q )∇ l k uD k J α2 β2 ξα2 0 ξβ2 0 + s 1 F i j R k i1j D k J α2 β2 ξα2 0 ξβ2 0 - 2r 1 [ ∂ 1 ρ 0 ρ 0 - (1 -t)∂ 1 ρ 0 + t∂ p ρ 1 ∂ 1 X p ρ t -B p q ∂ 1 D p X q + F i j S q i S j p ∂ 1 A p q ] + 2r 1 ( t∂ p ρ 1 D l X p ρ t + B p q D 2 pl X q - F i j S q i S j p D l A p q )∇ l 1 u -2r 2 1 F i j R 1 i1j }.
By the critical condition (5.35), Lemma 4.3, Lemma 4.4, Proposition 5.1, Cauchy-Schwarz inequality, we deduce that there exists a positive constant such that the following estimate holds:

V I 2 ≤ 1 r 2 + r 3 {CM + C(ε + δ)F 1 1 + [C(ε + δ) -π 2 ]F α1 α1 + C(ε + δ)F i j ∇ α1 i u∇ j α1 u + C(ε + δ)F n1+1 n1+1 + [C(ε + δ) -2π 2 ]F α2 α2 + C(ε + δ)F i j ∇ α2 i u∇ j α2 u} + 1 s 2 + s 3 {CM + C(ε + δ)F n1+1 n1+1 + [C(ε + δ) -π 2 ]F α2 α2 + C(ε + δ)F i j ∇ α2 i u∇ j α2 u + C(ε + δ)F 1 1 + [C(ε + δ) -2π 2 ]F α1 α1 + C(ε + δ)F i j ∇ α1 i u∇ j α1 u}. where M = max{max M1×M2 |d log ρ 0 |, max M1×M2 |d log ρ 1 |}.
Plugging the upper bounds (5.40),(5.41),(5.42),(5.44),(5.45),(5.47) into the inequality (5.36),we obtain the following inequality: If δ is small enough, we can assume max{r 2 , s 2 } < (π-δ) 2

0 ≤ 1 r 2 + r 3 {8 + CM + [4C(ε + δ) -2]F 1 1 + [3C(ε + δ) -π 2 ]F α1 α1 + [3C(ε + δ) + 2r 2 r 2 1 -1]F i j ∇ α1 i u∇ j α1 u + [C(ε + δ) -2]F n1+1 n1+1 + [C(ε + δ) -2π 2 ]F α2 α2 + [C(ε + δ) -2]F i j ∇ α2 i u∇ j α2 u} + 1 s 2 + s 3 {8 + CM + [4C(ε + δ) -2]F n1+1 n1+1 + [3C(ε + δ) -π 2 ]F α2 α2 + [3C(ε + δ) + 2s 2 s 2 1 -1]F i j ∇ α2 i u∇ j α2 u + [C(ε + δ) -2]F 1 1 + [C(ε + δ) -2π 2 ]F α1 α1 + [C(ε + δ) -2]F i j ∇ α1 i u∇ j α1 u}. Fixing ε < 1 24C , δ <

4

. Hence, we have

0 ≤ 8 + CM - 1 4 F i j S k i S j k .
By the inequality of arithmetic geometric means and the equation (5.46), we get that there exists a positive number C such that

F i j S k i S j k ≥ C(min{1, ρ 1 ρ 0 } 1 n 1 +n 2 1 (r 2 s 2 ) 1 n 1 +n 2 .
Proof. It is clear that left side is equal to trF when ν = 0. So we assume that ν ̸ = 0. Taking the Fermi coordinate system along the geodesic exp m (s ν |ν| ). If |ν| ≥ 3π 4 , set R(•) = R(•, ν)ν. In view of the curvature assumption (5.47) and the positive definiteness of the linear operator F, we have trSFS + trFR ≥ F 1 1 + trF R -εtrF

= F 1 1 + |ν| 2 n ∑ i=2 F i i -εtrF ≥ (1 -ε)trF ≥ 1 2 trF,
where the last inequality yields provided ε < 1 2 . If |ν| ≤ 3π 4 , the condition of Theorem 3.1 is satisfied provided ε <

1 3π √ 2(n-1)
. From (3.10), we know that

|J -1 0 -J-1 0 | ≤ 4 √ n -1( |ν| sin |ν| ) 2 ε ≤ 9 2 π 2 √ n -1ε, (5.48)
where the last inequality follows from the fact that the function t sin t is increasing in (0, π). Observe that S -S = (J -1 0 -J-1 0 )J 1 + J-1 0 (J 1 -J1 ).

By (5.48)

F i i + |ν| 2 n ∑ i=2 F i i -CεtrF = F 1 1 + |ν| 2 sin 2 |ν| n ∑ i=2 F i i -CεtrF ≥ (1 -Cε)trF,
The desired inequality follows if we choose ε < min{ 1 2C ,

1 3π √ 2(n-1)

}.

We now prove Theorem 5.2 by the continuity method. Assume that the condition of Theorem 5.2 is satisfied.

Let I be the set of the parameter t ∈ [0, 1] for which there exists a C k+2,α solution u t of the Monge-Ampère type equation (5.29) with ρ 1 replaced by ρ t = (1t)ρ 0 + tρ 1 .

To ensure the uniqueness, we assume that ∫ M1×M2 u t dvol = 0. It is clear that 0 ∈ I, so the set I is nonempty. The openness is derived by an implicit function theorem [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]. The connectedness of the set [0, 1] will prove the equation (5.29) admits a C k,α solution if I is closed.

In subsection 5.3.2, we have proved the estimate (5.2). By Lemma 5.1, in order to prove Theorem 5.2, it is sufficient to prove that there exists a positive constant C such that ∀t ∈ I, max From Corollary 3.1, we know that the MTW tensor on M 1 × M 2 satisfies the A3W condition but not A3S condition. Moreover, it is non-negative. Thus the condition of Theorem 6.1 in Delanoë [START_REF] Delanoë | On the smoothness of the potential function in Riemannian optimal transport[END_REF] is not satisfied. We have to prove the second derivative estimate. Proof of Theorem 5.2. Suppose that the condition of Theorem 5.2 is established. As mentioned before, it is sufficient to prove (5.49). Let

H = ∇ 2 u + ∇ 2 d 2 G 2 ,
where G is the optimal transport map and the derivative of It is clear that h attains the maximum at the point (m 0 , ξ 0 ). Take the coordinate system z = (x, y), where x is normal coordinate system in M 1 around the point p0 and y is normal coordinate system in M 2 around the point p0 . The associated coordinate system in tangent bundle T (M 1 × M 2 ) is denoted by (z, v), where v = v i ∂ ∂z i . Components of tensors will be denoted by:

∇ p u = ∇ i u(p) ∂ ∂z i , ∇ 2 p u = ∇ i j u(p)dz j ⊗ ∂ ∂z i , S = S i j (p, ν, 1)dz j ⊗ ∂ ∂z i , H = H i j (p)dz j ⊗ ∂ ∂z i , F = F i j (p)dz j ⊗ ∂ ∂z i ,
where H i k F k j = δ i j . Suppose that the tangent vector ξ 0 = ∂ ∂z 1 p0 . We may also assume that (H i j ) is diagonal at the point p 0 .

Let m be in the domain of the coordinate system z. The associated coordinate is given by z = (z 1 , • • • , z n1+n2 ). Fix ξ ∈ T m (M 1 × M 2 ). The associated coordinate of ξ is denoted by ξ = ξ i ∂ ∂z i . Then ⟨Hξ, ξ⟩ = H a b g ap ξ b ξ p , |ξ| 2 = g ab ξ a ξ b .

(5.50)

In the following all terms are evaluated at the point (p 0 , ξ 0 ). It will be implicitly understood throughout the calculations.

It is clear that the function log h also attains its maximum at the point (p 0 , ξ 0 ). The first derivative condition From (5.50), by differentiating the function log h with respect to z i , the first derivative condition could be read as:

∇ i H 1 1 H 1 1 + β∇ k i u∇ k u = 0.
(5.51)

The second derivative condition Differentiating twice the test function log h with respect to z i , z j , the second derivative condition can be written as follows:

0 ≥ I 3 + II 3 + III 3 , (5.52)
where

I 3 = 1 H 1 1 F i j ∇ j i ∇ 1 1 u - 1 (H 1 1 ) 2 F i j ∇ i H 1 1 ∇ j H 1 1 , II 3 = βF i j ∇ j ∇ k i u∇ k u + 1 H 1 1 F i j ∇ j i S 1 1 , III 3 = βF i j ∇ k i u∇ j k u + 1 H 1 1 F i j ∂ 2 ij g k1 H k 1 -F i j ∂ 2 ij g 11 .
We denote by I 3 , II 3 and III 3 these terms. Term I 3 The term I 3 involves the fourth derivative of the potential function u. Making use of the covariant derivative commutative formula

∇ klij u = ∇ ijkl u + (∇ j R s kli + ∇ l R s ikj )∇ s u + R sklj ∇ s i u + R skli ∇ s j u + R silj ∇ s k u + R sikj ∇ s l u,
we get:

F i j ∇ j i ∇ 1 1 u = F i j ∇ 1 1 ∇ j i u + F i j (∇ j R s 11i + ∇ 1 R s i1j )∇ s u + 2F i j (R s11j ∇ s i u + R si1j ∇ s 1 u).
Thus

I 3 = 1 H 1 1 (F i j ∇ 1 1 ∇ j i u - 1 H 1 1 F i j ∇ i H 1 1 ∇ j H 1 1 ) + 1 H 1 1 [F i j (∇ 1 R s i1j + ∇ j R s 11i )∇ s u + 2F i j (R si1j H s 1 + R s11j H s i ) - 2F i j (R si1j S s 1 + R s11j S s i )].
Differentiating the equation (5.29) twice, we derive

F i j ∇ 1 1 ∇ j i u = F i s ∇ 1 H s q F q j ∇ 1 H j i + ∇ 1 1 ϕ -F i j ∇ 1 1 S i j = F i i F j j (∇ 1 H j i ) 2 + ∇ 1 1 ϕ -F i j ∇ 1 1 S i j .
The first term in the above expression is non-negative. Moreover, it can control the negative term in I 3 . Indeed,

F i i F j j (∇ 1 H j i ) 2 - 1 H 1 1 F i i (∇ i H 1 1 ) 2 - 1 2H 1 1 ∑ i≥2 F i i (∇ i H 1 1 ) 2 = F i i F j j (∇ 1 H j i ) 2 - 1 (H 1 1 ) 2 (∇ 1 H 1 1 ) 2 - 3 2H 1 1 ∑ i≥2 F i i (∇ i H 1 1 ) 2 ≥ 1 2H 1 1 ∑ i≥2 F i i (∇ i H 1 1 ) 2 + 2 H 1 1 ∑ i≥2 F i i [(∇ 1 H 1 i ) 2 -(∇ i H 1 1 ) 2 ] ≥ - 6 H 1 1 ∑ i≥2 F i i (∇ 1 H 1 i -∇ i H 1 1 ) 2 .
where the last inequality follows from

2 H 1 1 ∑ i≥2 F i i [(∇ 1 H 1 i ) 2 -(∇ i H 1 1 ) 2 ] = 2 H 1 1 ∑ i≥2 F i i [∇ 1 H 1 i -∇ i H 1 1 ][∇ 1 H 1 i + ∇ i H 1 1 ] = 2 H 1 1 ∑ i≥2 F i i [∇ 1 H 1 i -∇ i H 1 1 ][∇ 1 H 1 i -∇ i H 1 1 + 2∇ i H 1 1 ] ≥ - 1 2H 1 1 ∑ i≥2 F i i (∇ i H 1 1 ) 2 - 6 H 1 1 ∑ i≥2 F i i (∇ 1 H 1 i -∇ i H 1 1 ) 2 .
As a consequence, we have

I 3 ≥ - 6 (H 1 1 ) 2 ∑ i≥2 F i i (∇ 1 H 1 i -∇ i H 1 1 ) 2 + 1 H 1 1 (∇ 1 1 ϕ -F i j ∇ 1 1 S i j ) + (5.53) 1 H 1 1 [F i j (∇ 1 R s i1j + ∇ j R s 11i )∇ s u + 2F i j (R si1j H s 1 + R s11j H s i ) - 2F i j (R si1j S s 1 + R s11j S s i )] = I 31 + I 32 + I 33 .
We first deal with I 31 . After commuting the third derivatives, we see that

∇ 1 H 1 i -∇ i H 1 1 = R k 1i1 ∇ k u + ∂ 1 S 1 i -∂ i S 1 1 + D k S 1 i ∇ k 1 u -D k S 1 1 ∇ k i u = R k 1i1 ∇ k u + ∂ 1 S 1 i -∂ i S 1 1 - D k S 1 i S k 1 + D k S 1 1 S k i + D k S 1 i H k 1 -D k S 1 1 H k i = R k 1i1 ∇ k u + ∂ 1 S 1 i -∂ i S 1 1 - D k S 1 i S k 1 + D k S 1 1 S k i + D 1 S 1 i H 1 1 -D i S 1 1 H i i ,
where the last equality follows from the fact that the matrix (H i j ) is diagonal at p 0 . Note that H 1 1 is the maximal eigenvalue of (H i j ). Assume that H 1 1 ≥ 1. From Lemma 5.2, we have

I 31 ≥ -C -CtrF.
(5.54)

We now treat I 32 . By a lengthy computation, we get

∇ 1 1 ϕ = ∂ 2 11 ϕ + 2∂ 1 D k ϕ∇ k 1 u + D 2 pq ϕ∇ p 1 u∇ q 1 u + D k ϕ(∂ 1 ∇ k 1 u -∂ 1 Γ k 1s ∇ s u).
After commuting the third derivative, we obtain

∇ 1 1 ϕ = ∂ 2 11 ϕ + 2∂ 1 D k ϕ∇ k 1 u + D 2 pq ϕ∇ p 1 u∇ q 1 u + D k ϕ(∂ k ∇ 1 1 u + R s 1k1 ∇ s u -∂ 1 Γ k 1s ∇ s u) = ∂ 2 11 ϕ + 2∂ 1 D k ϕ∇ k 1 u + D 2 pq ϕ∇ p 1 u∇ q 1 u + D k ϕ(∇ k H 1 1 + R s 1k1 ∇ s u -∂ k S 1 1 - D p S 1 1 ∇ p k u -∂ 1 Γ l 1s ∇ s u).
After commuting the derivatives, we find

II 3 = βF i j ∇ k ∇ j i u∇ k u + βF i j R l ikj ∇ l u∇ k u + 1 H 1 1 F i j (∂ 2 ij S 1 1 + 2∂ i D k S 1 1 ∇ k j u + D 2 kl S 1 1 ∇ k i u∇ l j u + D k S 1 1 ∇ k ∇ j i u + D k S 1 1 R s ikj ∇ s u -D k S 1 1 ∂ j Γ k iq ∇ q u) = βF i j ∇ k ∇ j i u∇ k u + βF i j R l ikj ∇ l u∇ k u + 1 H 1 1 F i j ∇ k ∇ j i uD k S 1 1 + 1 H 1 1 F i j (∂ 2 ij S 1 1 + 2∂ i D k S 1 1 ∇ k j u + D 2 kl S 1 1 ∇ k i u∇ l k u + D k S 1 1 R s ikj ∇ s u -D k S 1 1 ∂ j Γ l iq ∇ q u).
Differentiating the equation (5.29) with respect to z k , we deduce

F i j ∇ k ∇ j i u = ∂ k ϕ + D l ϕ∇ l k u -F i j ∂ k S i j -F i j D l S i j ∇ l k u.
Thus

II 3 = β(∂ k ϕ + D l ϕ∇ l k u -F i j ∂ k S i j -F i j D l S i j ∇ l k u)∇ k u + βF i j R l ikj ∇ l u∇ k u + 1 H 1 1 D k S 1 1 (∂ k ϕ + D q ϕ∇ q k u - F i j ∂ k S j i -F i j D q S j i ∇ q k u) + 1 H 1 1 F i j (∂ 2 ij S 1 1 + 2∂ i D k S 1 1 ∇ k j u + D 2 kl S 1 1 ∇ k i u∇ j l u + D k S 1 1 R s ikj ∇ s u -D k S 1 1 ∂ j Γ l iq ∇ q u).
Let us observe that

F i j ∇ k i u = δ k j -F i j S k i ,
(5.60)

F i j ∇ k i u∇ j l u = H k l -2S k l + F i j S k i S j l .
(5.61)

Together with Lemma 5.2, we infer

II 3 ≥ -C -C(1 + β)∥ϕ∥ C 1 -CtrF + (5.62) βD l ϕ∇ l k u∇ k u -βF i j ∂ k S i j ∇ k u - βF i j D l S j i ∇ l k u∇ k u + βF i j R l ikj ∇ l u∇ k u.
The term III 3 In view of (5.61), we get

III 3 = βtrH -2βtrS + βtrSFS + 1 H 1 1 F i j ∂ 2 ij g k1 H k 1 -F i j ∂ 2 ij g 11 ≥ -Cβ -CtrF + βH 1 1 + βtrSFS. ( 5

Perspectives

In this thesis, we mainly prove the smoothness of the optimal transport map on two classes of compact Riemannian manifold which are nearly spherical manifolds and Riemannian products of nearly spherical manifolds. It is interesting to find other manifolds such that the corresponding optimal transport map is smooth.

We list some open questions about the optimal transportation in references.

(1) Does the A3W condition imply that the injectivity domain is convex?

Loeper-Villani [START_REF] Loeper | Regularity of optimal transport in curved geometry: the nonfocal case[END_REF] showed that the A3S condition implied that injectivity domain is uniformly convex in case of non-focal Riemannian manifold. Figalli-Gallouët-Rifford [START_REF] Figalli | On the Convexity of Injectivity Domains on Nonfocal Manifolds[END_REF] showed that the A3W condition deduced that injectivity domain is convex in case of non-focal Riemannian manifold.

(2) Is the continuity of the optimal transport map equivalent to the A3W condition and the convexity of the injectivity domain?

Figalli-Rifford-Villani [START_REF] Figalli | Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds[END_REF] proved that the A3W condition and the convexity of the injectivity domain is necessary for the continuity of the optimal transport map and also sufficient in dimension 2.

(3) Does one has the control on the Hausdorff dimension of the singular set?

Figalli [START_REF] Figalli | Regularity Properties of Optimal Maps Between Nonconvex Domains in the Plane[END_REF] proved that the singular set is a 1-dimensional manifold of class C 1 out of a countable set in the plane when the target is not convex. But the result is not known in high dimension.

S

  0 , µ 1 ) = {γ : γ est une mesure de probabilité sur X × Y. Pour les applications πX : X × Y → X, (x, y) → x et π Y : X × Y → X, (x, y) → y, on a (π X ) # γ = µ 0 , (π Y ) # γ = µ 1 }.La mesure µ est appelée le couplage optimal. Ce problème de minimisation est plus général. L'existence est connue sous certaines hypothèses. Par exemple, soient X ix x CONTENTS et Y deux espaces métriques complets séparables. Soit c une fonction semi-continue inférieure. Le couplage optimal existe [98] [87]. Kantorovich a montré que le problème (1) a une formation duale. Concrètement, on cherche une paire de fonctions (u, v) telle que -= {(ϕ, ψ) : ∀(x, y) ∈ X × Y, ϕ(x) + ψ(y) ≥ -c(x, y)}. Lorsque le maximum est atteint par (u, v), on a u(x) = sup y∈Y [-c(x, y)v(y)], v(y) = sup x∈X [-c(x, y)u(x)].

  On donne des exemples des fonctions c-convexes. Lorsque X = Y = R n et c(x, y) = -⟨x, y⟩, la c-convexité est équivalente à la convexité usuelle. Lorsque X = Y = R n et c(x, y) = 1 2 |x -y| 2 , une fonction u est c-convexe si et seulement si la fonction u + 1 2 |x| 2 est convexe. Lorsque X = Y = M un espace métrique et c(x, y) = d(x, y), la fonction u est c-convexe si et seulement si |u(x)u(y)| ≤ c(x, y), ∀x, y ∈ M. Il est clair que la fonction u est c-convexe si et seulement s'il existe une fonction v : Y → R telle que u(x) = sup y∈Y [-c(x, y)v(y)]. De plus, pour une solution du probléme (2), des fonctions u et v sont c-convexes.

0. 3 .

 3 LA R ÉGULARIT É DE L'APPLICATION DU TRANSPORT OPTIMAL xi 0.3 La régularité de l'application du transport optimal

  d et dvol désignent respectivement la distance géodésique et l'élément de volume sur M. Etant donné m ∈ M , Cut(m) note le lieu de coupure du point m. Le domaine d'injectivité du point m est noté par I(m). Définition 0.2. (Le tenseur de MTW) Soient m ∈ M, ν ∈ I(m), ξ, η ∈ T m M . Le tenseur de MTW est défini par C(m, ν)(ξ, η) = -

  ce qui donne C(m, 0)(ξ, η) = R m (ξ, η, ξ, η). (4) 0.3. LA R ÉGULARIT É DE L'APPLICATION DU TRANSPORT OPTIMAL xiii

Définition 0. 3 .

 3 (La condition de MTW) (i) On dit que le tenseur de MTW est positif, si pour tout m ∈ M, ξ, η ∈ T m M, on a C(m, ν)(ξ, η) ≥ 0. (ii) On dit que la condition A3W est vérifiée, si pour tout m ∈ M et pour tous ξ, η ∈ T m M avec g m (ξ, η) = 0, on a C(m, ν)(ξ, η) ≥ 0. (iii) On dit que la condition A3S est vérifiée, si pour tout m ∈ M et pour tous ξ, η ∈ T m M \ {0} avec g m (ξ, η) = 0, on a C(m, ν)(ξ, η) > 0.

  Recall that the cut time is smaller than the focal time. The injectivity domain is an open subset contains the origin in T m M and star-shaped with respect to the origin. Moreover,M = exp m (I(m)) ⊔ Cut m , where ⊔ means disjoint union. The exponential map exp m : I(m) → M \Cut m is a diffeomorphism.

  are all bounded above by C 1 .

Remark 1 . 2 .

 12 For later use, regarding |J 0 -J0 |, the constant C 2 can be taken value 2 √ n -1.(to see Remark 5 of [29])

2 1 .

 1 Plugging (3.35)(3.27)(3.34) and (3.26)into the above expression, we get

2 ,

 2 Lemma 4.3 and Lemma 4.4, and combining with the facts

2 ,

 2 Lemma 4.3 and Lemma 4.4,Lemma 4.5, Lemma 4.6, together with the facts

Lemma 5 . 2 .

 52 Let (M, g) be a compact Riemannian manifold of dimension n ≥ 2. The diameter of (M, g) is denoted by D. Then for all admissible function u with ∫ M udvol = 0, the following estimates hold max M |u| ≤ D 2 , max M |∇u| ≤ D. Proof. Let the function u be admissible with ∫ M udvol = 0. Fix m ∈ M. By definition of the admissible function, the curve exp m (t∇ m u) is a minimizing geodesic from m to exp m ∇ m u. Thus |∇ m u| = d(m, exp m ∇ m u) ≤ D.

Claim 5 . 1 .

 51 Set J ( m, m, ν, ν) = J ( m, ν)+ J ( m, ν) where J ( m, ν) = -|ν| 2 S -1 ( m, ν, 1) and J ( m, ν) = -|ν| 2 S -1 ( m, ν, 1). Then the minimum min{(⟨J ξ, ξ⟩ + π 2 -| ∇ mu| 2 )(⟨J ξ, ξ⟩ + π 2 -| ∇ mu| 2 ) :

  u = √ |g|(z)ρ 0 (z) √ |g|(Z) det(D v Z)ρ t (z) . (5.46)

m∈M1×M2 |∇ 2

 2 m u t + S(m, ∇ m u t , 1)| ≤ C.(5.49)

d 2 p 2 2 |∇u| 2 m

 2222 (m) = d 2 2 (m, p) is with respect to m and then to take value at p = G(m).Consider the maximization problem max{⟨Hξ, ξ⟩eβ : m ∈ M 1 × M 2 , ξ ∈ T m (M 1 × M 2 ), |ξ| m = 1},where β is a positive constant to be determined later.Assume that the maximum is achieved at the point (p0 , ξ 0 ). Fixing m ∈ M 1 ×M 2 , ξ ∈ T m (M 1 × M 2 )\{0},we consider the test function: h (m, ξ) = ⟨Hξ, ξ⟩ |ξ| 2 e β 2 |∇u| 2 m .

. 63 )

 63 Substituting (5.59)(5.62)(5.63) into (5.52), we see that0 ≥ -C(1 + β) -C(2 + β)∥ϕ∥ -3CtrF + (β -C∥ϕ∥)H 1 1 -βF i j ∂ k S i j ∇ k u + β(trSFS + trFR).where R(•) = R(•, ∇u)∇u.Note that Theorem 4.1 and (3.10) imply that there exists a positive constant Λ 1 depending only on n 1 , n 2 such that |F∂ x S| ≤ Λ 1 ε max{trF, trSFS}.

  le potentiel du transport optimale est de classe C k+2,α pour des mesures de probabilité sur M ρ 0 dvol et ρ 1 dvol à densité strictement positive de classe C k,α .

	Le Théorème 0.2 implique que sur une variété presque sphérique lisse, si la densité
	des mesures est régulière et strictement positive, alors l'application du transport optimal
	est régulière.

Corollaire 0.2. Sous les mêmes hypothèses que le théorème 0.2, suppose que les densités ρ 0 , ρ 1 sont de classe C ∞ . Alors l'application du transport optimal est de classe C ∞ .

  The Ricci tensor is obtained by the contraction Ric ij = g kl R ikjl and the scalar curvature by Scal = g ij Ric ij .

, • • • , x n }, the components of Riemann curvature tensor are given by R( ∂ ∂x j , ∂ ∂x k ) ∂ ∂x i = R l ijk ∂ ∂x l and R ijkl = g ip R p jkl respectively.

  .24) where |ξ ∧ η| 2 m , |η ∧ ν| 2 m and |ξ ∧ ν| 2 m stand for the squared areas of the parallelograms defined in T m M ,i.e.: |ξ ∧η| 2 m

  .[START_REF] Figalli | An approximation lemma about the cut locus, with applications in optimal transport theory[END_REF] Under the above estimations, we can imply the lower bound of the M T W tensor.

	Assume that 0 < ε < ε3 2C , 0 < τ < min{ √ ε3 2C , δ3 } with ε 3 small enough.
	Substituting the lower bounds (3.47),(3.48) and (3.49) into (3.47), we can derive:

  Let E a (t) be the matrix valued function whose elements E i

	ab (t) are given by the ab . Combining the equations (4.1) and (4.2) with (1.15),(1.16), Lemma ab -Ji difference J i
	4.1, Lemma 4.2 and Lemma 4.3, we find that there exists a positive constant C > 0
	such that | Ëa + RE a | ≤ Cε. Applying a representation formula [29] to E i ab (t), it yields
	|E a | ≤ Cε. The remaining approximations come from the representation:
	∫ t
	Ėi ab (t) =	Ëi ab (s)ds.
	0

  Then by the Bishop's theorem, det d ∇mut exp m is uniformly bounded from above by 1 if M has non-negative Ricci curvature. As mentioned in Section 1.2.1, we know that det d ∇mut exp m is positive. But det d ∇mut exp m may not has a positive lower bound. Recall det J 0 (m, ν, 1) vanishes if (and only if) exp m ν is conjugate to m. Hence, the estimate (5.2) is not obvious. For instance, on the round sphere S n , det d ∇mut exp m = ( sin |∇mut| |∇mut| ) n-1 is close to zero as |∇ m u t | approaches π. Observe that the assumption (1.3) infers that the length of gradient ∇ m u t is strictly less than π. Making use of Lemma 1.2, the estimate (5.2) is obvious if max{|∇ m u t | : m ∈ M } ≤ 3π

  .38) Note that -( S α1 β1 ) and -( S α2 β2 ) are both uniformly bounded from below under the curvature assumption (1.3). In fact, from the Hessian Comparison Theorem, we know that -( S α1 β1 ) ≥ -r1 cos r1 sin r1 I n-1 and -( S α2 β2 ) ≥ -s1 cos s1 sin s1 I n-1 . Making use of the fact that the real function -t cos t sin t is increasing in ( π 2 , π), for min{r 1 , s 1 } ≥ 3π 4 , we have We will cope with I 2 to V I 2 term by term. The term I 2 It is obvious that I 2 is non-positive. That is

	-( S α1 β1 ) ≥	3π 4	I n-1 , -( S α2 β2 ) ≥	3π 4	I n-1 .	(5.39)
			I 2 ≤ 0.			(5.40)

Remerciements

Chapter 6. Perspectives Bibliography

Ceci provient du fait que l'on n'a

Chapter 5

The smoothness of the optimal transport map This chapter is concerned with the smoothness of the optimal transport map on two classes of compact Riemannian manifolds which are nearly spherical manifolds and Riemannian products of nearly spherical manifolds. The optimal transport map is given by exp(grad u), where the potential function u satisfies a Monge-Ampère type equation. By the method of maximum principle, we prove that the Jacobian of the exponential map at grad u has an uniform positive lower bound. Then the Ma-Trudinger-Wang's device, together with the method of continuity implies the smoothness of the optimal transport map.

Introduction

Let (M, g) be a closed Riemannian manifold of dimension n ≥ 2. Assume that (M, g) is smooth.

Let Hess (c) u be c-Hessian of u, namely,

As in subsection 0.3.1, the C 2 potential function u of the optimal transport map G(m) = exp m ∇ m u, pushing forward ρ 0 dvol to ρ 1 dvol, satisfies the following Monge-Ampère type equation: det(d ∇mu exp m ) det Hess (c)

Conversely, a classical C 2 solution of the above equation is the potential function of the optimal transport map G pushing forward ρ 0 dvol to ρ 1 dvol. We say that a C 2 function u : M → R is admissible if for every point m ∈ M, ∇ m u ∈ I(m) and Hess (c) m u > 0. It is known that the C 2 solutions of (5.1) are unique up to a constant(see [START_REF] Delanoë | On the smoothness of the potential function in Riemannian optimal transport[END_REF]). We consider the regularity of the potential function u, that is, given (k, α) ∈ N × (0, 1), with k ≥ 2, if both ρ 0 and ρ 1 are C k,α , we want to know whether the solution u is C k+2,α .

We will address the above problem by the continuity method. Let I be the set of the parameter t ∈ [0, 1] for which there exists a C k+2,α admissible solution u t of the equation (5.1) by replacing ρ 1 by ρ

To ensure the uniqueness, the solutions u t are normalized by ∫ M u t dvol = 0. It is clear that 0 ∈ I, so the set I is nonempty. The openness is derived by an implicit function theorem [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]. The connectedness of the interval [0, 1] will imply that the equation (5.1) admits a C k,α solution if I is closed. 55

Preliminary

In this subsection, we recall some facts about product Riemannian manifolds. Let (M 1 , g) and (M 2 , ĝ) be two complete smooth Riemannian manifolds of dimension n 1 ≥ 2 and n 2 ≥ 2 respectively. It is well known that the tangent vector space has the decomposition

Denote by (M 1 × M 2 , g) the Riemannian product of (M 1 , g) and (M 2 , ĝ). We denote by ∼ and ∧ the projection mappings of T (M 1 × M 2 ) to T M 1 and T M 2 respectively. For vector fields

where ∇, ∇ denote the Levi-Civita connections of M 1 , M 2 respectively. In addition, the Riemannian curvature tensor Riem of

where Riem, Riem denote the Riemannian curvature tensors of M 1 , M 2 respectively. Let γ(t) = ( γ(t), γ(t)) be a curve on

The Jacobi matrices with the initial conditions on Riemannian product M 1 × M 2 can be described by ones on M 1 and M 2 . More precisely, we have the following result.

) is an orthonormal parallel transport moving frame along γ. Moreover, the Jacobi matrices with the initial conditions J 0 and J 1 takes the form:

where Ja and Ĵa are the Jacobi matrices with the initial conditions on M 1 and M 2 respectively.

Proof. The Lemma follows from (1.7), (1.8) and the decomposition (5.31).

As a direct consequence, the Hessian of squared distance on M 1 × M 2 can be decomposed into the Hessian of squared distance on M 1 and M 2 .

Corallary 5.3. Suppose the same assumptions as in Lemma 5.4

ν, t) be the linear operator from T mM 1 to T mM 1 whose matrix in the orthonormal basis {ẽ 1 (0), ẽ2 (0), • • • , ẽn1 (0)} is given by t J0 (t) -1 J1 (t) and S( m, ν, t) be the linear operator from T mM 2 to T mM 2 whose matrix in the orthonormal basis {ê 1 (0), ê2 (0),

where S(m, ν, t) is the linear operator from

Thus r 2 s 2 is bounded from below by

This ends the proof of the Claim 5.1. Proof of estimate (5.2). We will prove the estimate (5.2) in three steps. Fix ( m, m) ∈ M 1 × M 2 .

Step 1.We first treat the case:| ∇ mu| ≤ η 1 , | ∇ mu| ≤ η1 for some constants η1 , η1 ∈ (0, π). Using Lemma 1.2 and the definition of determinant, there exists a positive constant C > 0 such that

Recall that real function sin s s is increasing in the interval (0, π). By choosing ε small enough, we get the estimate.

Step 2.In this step, we will examine the case:min{| ∇ mu|, | ∇ mu|} > πδ, for δ ∈ (0, π 4 ) small enough. Using Claim 5.1, by choosing δ small enough, we derive that both the second eigenvalue of J and the second eigenvalue of J have some positive lower bounds. By definition of J and J , we obtain that both det J0 and det Ĵ0 have positive lower bounds.

Step 3.In the last step, we address the case

Without loss of generality, we assume that | ∇ mu| ≥ π-η 0 , | ∇ mu| ≤ η2 for some constants η 0 ∈ (0, π 4 ), η2 ∈ (0, π). The constant η 0 will be determined later. As same as step 1, for | ∇ mu| ≤ η2 , the determinant det Ĵ0 ( m, ∇ mu, 1) has a positive uniform lower bound. To complete the proof, it suffices to show that the determinant det J0 ( m, ∇ mu, 1) also has a positive lower bound.

Note that one can prove that the minimization problem min{⟨ J ξ, ξ⟩

4 } also has a positive lower bound. Thus as same as step 2, det Ĵ0 ( m, ∇ mu, 1) has a positive uniform lower bound. This completes the proof of the estimate (5.2).

Proof of Theorem 5.2

This subsection is devoted to the proof of Theorem 5.2. It is known that the sectional curvatures of M 1 × M 2 are non-negative and may vanish, besides its c-curvature is nonnegative and may vanish on some directions. The vanishing of c-curvature are the main obstacles of the smoothness. We will derive the uniformly C 2 estimate by estimate (5.2). Then the method of continuity implies the smoothness of the optimal transport map.

To begin with, we give a basic lemma.

Lemma 5.5. Let (M, g) be a closed Riemannian manifold of dimension n ≥ 2. Suppose that (M, g) satisfies (1.3). Then there exists a positive constant Λ 0 depending only on n such that, for where

Making use of the critical condition (5.51) again, it follows that

In view of Lemma 5.2, we derive

Similarly,

Note that the MTW tensor on M 1 × M 2 is non-negative. Thus

By virtue of (5.55) and (5.56), it follows that

For I 33 , using Lemma 5.2, we get

(5.58) Substituting (5.54)(5.57)(5.58) into (5.53), we see that

The term II 3 We are in position to deal with the term II 3 . It is readily to see that

Choosing 0 < ε < Λ 0 , due to Lemma 5.5, we derive

Taking ε < 1 8πΛ1 and β ≥ max{2C∥ϕ∥, 24C}, we get ∥ϕ∥H 1 1

(5.64)

Thus the H 1 1 is bounded above at the point p 0 . By the positivity of H, thus |H| is bounded from above. This finishes the proof of Theorem 5.2.

The smoothness on C 4 perturbation of product Riemannian manifold

In this section, we prove Theorem 5.3. Let (M 1 , g) and (M 2 , ĝ) be two closed Riemannian manifolds of dimension n 1 ≥ 2 and n 2 ≥ 2 respectively. Let ρ 0 dvol and ρ 1 dvol be two smooth positive Borel probability measures on M 1 × M 2 . Set (M 1 × M 2 , g × ) be Riemannian product of (M 1 , g) and (M 2 , ĝ) and Riem the corresponding (4,0)-th Riemann curvature tensor.

Note that Theorem 5.3 is trivial if the optimal transport map on (M 1 × M 2 , g × ) is not smooth. Indeed, we just take g = g × . Without generality, assume that the optimal transport map on (M 1 × M 2 , g × ) is smooth.

Let h be a non-trivial C 4 smooth function on M 1 . Consider the conformal metric (M 1 × M 2 , g = e -2u g × ) with u = - 1 2 log(1 + εh 2 ). Note that u can be viewed as a function on

It is clear that g is C 4 perturbation of g × for ε sufficiently small, i.e.

∥gg × ∥ C 4 < ε, ε sufficiently small.

Let m0 ∈ M 1 be a point such that ∆u( m0 ) < 0. The existence of m0 follows from the method of integration by parts. Indeed, if ∆u is non-negative on M 1 . From the integration by parts, we know that ∫

Thus h is trivial, this gives the contradiction. Fix a point m0 ∈ M 2 . Let z = (x, y) be a local coordinate system where x is the geodesic normal coordinate system in M 1 centered at m0 and y is the geodesic normal coordinate system in M 2 centered at m0 .

It is known that the Riemann curvature tensor Riem u of the conformal metric g is given by It is known that the A3W condition is a necessary condition of the continuity of the optimal transport map. Note that the A3W condition implies that the sectional curvature on plane which is spanned by the mutually perpendicular vector is nonnegative. Using (5.65), we know that there exists some optimal transport map on (M 1 × M 2 , e -2u g × ) with the regular positive probability measures is not continuous.

In conclusion, we derive Theorem 5.3.