Abdelghani Zeghib

ON DISCRETE PROJECTIVE TRANSFORMATION GROUPS OF RIEMANNIAN MANIFOLDS

We prove rigidity facts for groups acting on pseudo-Riemannian manifolds by preserving unparameterized geodesics.

RÉSUMÉ. Nous démontrons des résultats de rigidité pour les groupes agissant sur des variétés pseudo-riemanniennes en préservant leurs géodésiques non-paramétrées.

1. INTRODUCTION 1.0.1. The projective group of a connection. Two linear connections ∇ and ∇ ′ on a manifold M are equal iff they have the same (parameterized) geodesics. They are called projectively equivalent if they have the same unparameterized geodesics. This is equivalent to that the difference (2, 1)-tensor T = ∇ -∇ ′ being trace free in a natural sense [START_REF] Eastwood | Notes on projective differential geometry, Symmetries and Overdetermined Systems of Partial Differential Equations[END_REF].

The affine group Aff(M, ∇) is that of transformations preserving ∇ and the projective one Proj(M, ∇) is that of transformations f sending ∇ to a projectively equivalent one. So, elements of Aff are those preserving (parameterized) geodesics and those of Proj preserve unparameterized geodesics.

Obviously Aff ⊂ Proj; and it is natural to look for special connections for which this inclusion is proper, that is when projective non-affine transformations exist? 1.0.2. Case of Levi-Civita connections. Let now g be a Riemannian metric on M and ∇ its Levi-Civita connection. The affine and projective groups Aff(M, g) and Proj(M, g) are those associated to ∇.

More generally, g and g ′ are projectively equivalent if so is the case for their associated Levi-Civita connections. This defines an equivalence relation on the space Riem(M) of Riemannian metrics on M. Let P (M, g) denote the class of g, i.e. the set of metrics shearing the same unparameterized geodesics with g. It contains R + .g, the set of constant multiples of g. Generically, P (M, g) = R + .g.

One crucial fact here is that P (M, g) is always a finite dimensional manifold whose dimension is called the degree of projective mobility of g. (This contrasts with the case of projective equivalence classes of connections which are infinitely dimensional affine spaces. Similarly, conformal classes of metrics are identified to spaces of positive functions on the manifold). It is actually one culminate fact of projective differential geometry to identify P (M, g) to an open subset of a finite dimensional linear sub-space L(M,g) of endomorphisms of T M (see §3). Being projectively equivalent for connections is a linear condition, but this is no longer linear for metrics (say because the correspondence g → its Levi-Civita connection, is far from being linear!). The trick is to perform a transform leading to a linear equation, see [START_REF] Bolsinov | Geometrical interpretation of Benenti systems[END_REF] for a nice exposition.

1.0.3. Philosophy. The idea behind our approach here is to let a diffeomorphism f on a differentiable manifold M act on the space Riem(M) of Riemannian metrics on M. That this action has a fixed point means exactly that f is an isometry for some Riemannian metric on M. One then naturally wonder what is the counterpart of the fact that the f -action preserves some (finite dimensional) manifold V ⊂ Riem(M). A classical similar idea is to let the isotopy class of a diffeomorphism on a surface act on its Teichmuller space [START_REF] Thurston | Three dimensional geometry and topology[END_REF]. Here, as it will be seen bellow, we are specially concerned with the case dimV = 2.

1.0.4. More general pseudo-Riemannian framework. All this generalizes to the pseudo-Riemannian case. One fashion to unify all is to generalize all this to the wider framework of second order ordinary differential equations (e.g. hamiltonian systems) on M, by letting their solutions playing the role of (parameterized) geodesics.

1.0.5. Rigidity of the projective group. We are interested here in a (very) natural and classical problem in differential geometry: Characterize pseudo-Riemannian manifolds (M, g) for which Proj(M, g) Aff(M, g), that is M admits an essential projective transformation? Constructing upon a long research history by many people (see for instance [START_REF] Matveev | Proof of the projective Lichnerowicz-Obata conjecture[END_REF][START_REF] Matveev | Pierre Gallot-Tanno theorem for closed incomplete pseudo-Riemannian manifolds and applications[END_REF]), we dare formulate more precisely:

Projective Lichnerowicz conjecture 1.1. Let (M, g) be a compact pseudo-Riemannian manifold. Then, unless (M, g) is a finite quotient of the standard Riemannian sphere, Proj(M, g)/Aff(M, g) is finite.

-Same question when compactness is replaced by completeness (this does not contain the first case since general pseudo-Riemannian non-Riemannian compact manifolds may be non-complete).

1.0.6. Gromov's vague conjecture. It states that rigid geometric structures having a large automorphism group, are classifiable [START_REF] D'ambra | Lectures on transformation groups: geometry and dynamics. Surveys in differential geometry[END_REF][START_REF] Gromov | Rigid transformations groups. Géométrie différentielle[END_REF]! This needs a precise experimental (realistic) formulation for each geometric structure. Our question above is an optimistic formulation in the case of metric projective connections (those which are of Levi-Civita type). The historical case was that of Riemannian conformal structures with a precise formulation, as in the projective case above with the sphere playing a central role, is generally attributed to Lichnerowicz, and solved by J. Ferrand [START_REF] Ferrand | The action of conformal transformations on a Riemannian manifold[END_REF][START_REF] Obata | The conjectures on conformal transformations of Riemannian manifolds[END_REF]. In the general conformal pseudo-Riemannian case, there are many "Einstein universes", i.e. conformally flat examples with an essential conformal group. A Lichnerowicz type conjecture would be that all pseudo-Riemannian manifolds with an essential conformal group are conformally flat. However, this was recently invalidated by C. Frances [START_REF] Frances | About pseudo-Riemannian Lichnerowicz conjecture[END_REF]. In the projective case, there is no natural candidate of a compact pseudo-Riemannian (non-Riemannian) manifold playing the role Einstein universes; it becomes a natural challenge to prove that indeed Proj/Aff is always finite in this situation?

In the vein of this vague conjecture, it is surely interesting to see to automorphism groups of non-metric projective connections... 1.1. Results. This very classical subject of differential geometry was specially investigated by the Italian and next the Soviet schools. All famous names: Beltrami, Dini, Fubini, Levi-Civita are still involved in results on projective equivalence of metrics [START_REF] Beltrami | Risoluzione del problema: riportare i punti di una superficie sopra un piano in modo che le linee geodetische vengano rappresentante da linee rette[END_REF][START_REF] Dini | Sopra un problema che si presenta nella theoria generale delle rappresetazioni geografice di una superficie su un'altra[END_REF][START_REF] Levi-Civita | Sulle trasformazioni delle equazioni dinamiche[END_REF]. As for the "Soviet" side, let us quote o [START_REF] Aminova | Pseudo-Riemannian manifolds with general geodesics[END_REF][START_REF] Aminova | Projective transformations of pseudo-Riemannian manifolds[END_REF][START_REF] Sinjukov | Geodesic mappings of Riemannian spaces[END_REF][START_REF] Matveev | Proof of the projective Lichnerowicz-Obata conjecture[END_REF][START_REF] Matveev | Pierre Gallot-Tanno theorem for closed incomplete pseudo-Riemannian manifolds and applications[END_REF][START_REF] Solodovnikov | Projective transformations of Riemannian spaces[END_REF][START_REF] Solodovnikov | Geometric description of all possible representations of a Riemannian metric in Levi-Civita form[END_REF], and as names Solodovnikov who "introduced" the projective group problem, and last V. Matveev, who handled many remarkable cases of it.

1.1.1. Killing fields variant. Actually, it was Proj 0 (M, g), the identity component of Proj(M, g), that got real interest in the literature. Its elements are those belonging to flows of projective Killing fields. There is a prompt formulation of the Lichnerowicz conjecture here: if Proj 0 (M, g) Aff 0 (M, g), then (M, g) is covered by the standard sphere (assuming M compact).

This identity component variant was proved by V. Matveev in the case of Riemannian manifolds [START_REF] Matveev | Proof of the projective Lichnerowicz-Obata conjecture[END_REF], and remains open in the case of higher signature.

Local actions, i.e. projective Killing fields with flows defined only locally, were also considered, see for instance [START_REF] Bryant | A solution of a problem of Sophus Lie: normal forms of two-dimensional metrics admitting two projective vector fields[END_REF].

However, situations with no Killing fields involved, say for example when Proj is a discrete group do not seem to be studied. We think it is worthwhile to consider them because the discrete part may have dynamics stronger than the connected one, as in the case of a flat torus T n , but in fact for its affine group whose discrete part is the beautiful arithmetic (the best!) group SL n (Z).

1.1.2. Non dynamical variant. Without actions, one may think of having big P (M, g) as an index of symmetry, and one naturally may ask when this happens. For this, as in the projective case, consider A(M,g), the set of metrics affinely equivalent to g (i.e. having the same Levi-Civita connection). Here, we have the following wonderful theorem: Theorem 1.2 (Kiosak, Matveev, Mounoud, see [START_REF] Matveev | Pierre Gallot-Tanno theorem for closed incomplete pseudo-Riemannian manifolds and applications[END_REF]). Let (M, g) be a compact pseudo-Riemannian manifold. If dim P (M, g) ≥ 3, then P (M, g)= A(M,g), unless (M, g) is covered by the standard Riemannian sphere. In particular Proj(M, g) = Aff(M, g) in this case.

1.1.3. Rank 1 case? In view of this, it remains to consider the case dim P (M, g) = 2 (the dimension 1 case is trivial). Actually, this case occupies a large part in proofs of Lichnerowicz conjecture in the Riemannian as well as Kählerian cases [START_REF] Matveev | Proof of the projective Lichnerowicz-Obata conjecture[END_REF][START_REF] Matveev | Proof of the Yano-Obata conjecture for h-projective transformations[END_REF][START_REF] Matveev | Lichnerowicz-Obata conjecture in dimension two[END_REF]. (We think our approach here, besides it treats the discrete case, also simplifies these existing proofs). We are not surprized of the resistance of this case, reminiscent to a rank 1 phenomena, vs the higher rank case. Assuming dim P (M, g) ≥ 3 hides a symmetry abundance hypothesis! Anyway, in all our proofs, we will assume dim P (M, g) = 2.

1.1.4. Aim. Our first objective here is to provide a proof of the above conjecture in case of compact Riemannian manifolds Theorem 1.3. Let (M, g) be a compact Riemannian manifold. If M is not a Riemannian finite quotient of a standard sphere, and Proj(M, g) Aff(M, g), then Proj(M, g) is a finite extension of Aff(M, g). More precisely, Aff(M, g) = Iso(M, g), and a subgroup Iso ′ (M, g) of index ≤ 2, is normal in Proj(M, g), and the quotient group Proj(M, g)/Iso ′ (M, g) is either cyclic of order ≤ dim M, or dihedral of order ≤ 2 dim M.

Examples. In order to illustrate the non-linear character of projective equivalence, let us recall the Dini's classical result: two metrics on a surface are projectively equivalent, iff, at a generic point, they have the following forms in some coordinate system:

g = (X (x) -Y (y)(dx 2 + dy 2 ), ḡ = ( 1
Y (y) - 1 X (x) )( dx 2 X (x) + dy 2 Y (y) ) It follows that for the metric g = (a(x)-1 a(y) )( a(x)dx 2 + 1 √ a(y)
dy 2 ), the involution (x, y) → (y, x) is projective. This example given by V. Matveev [START_REF] Matveev | Proof of the projective Lichnerowicz-Obata conjecture[END_REF], shows that non-affine projective transformations may exist (outside the case of spheres) but are not in the identity component, because of his result. Theorem 1.3 says that the "discrete projective transformation group" is always finite, but we do not know examples more complicated than the last involution.

Remark 1.4. Some of quoted results are also true in the complete non-compact case, but we consider here compact manifolds, only.

1.2. Kähler version. Let (M, g) be a Hermitian manifold. Let V ⊂ M be a geodesic surface which is at the same time a holomorphic curve. If g is Kähler, then any (real) curve c in V satisfies that its complexified tangent direction is parallel; it is therefore called hplanar. It is very special that such V exists, but h-planer curves always exist. Two Kähler metrics are h-projectively equivalent if they share the same h-planer curves. A holomorphic diffeomorphism f is h-projective if f * g is h-projectively equivalent to g. Their group is denoted Proj Hol (M, g). (There exist equivalent terminologies for h-projective, as holomorphicprojective or c-projective). This holomorphic side of the projective transformation problem was classically investigated by the Japanese school [START_REF] Hasegawa | On holomorphically projective transformations of Kaehlerian manifolds[END_REF][START_REF] Ishihara | Holomorphically projective changes and their groups in an almost complex manifold[END_REF][START_REF] Yano | Isometry of Kaehlerian manifolds to complex projective spaces[END_REF][START_REF] Yoshimatsu | H-projective connections and H-projective transformations[END_REF].

Finally, V. Matveev and S. Rosemann generalize all known Riemannian results (on the identity component) to the Kähler case [START_REF] Matveev | Proof of the Yano-Obata conjecture for h-projective transformations[END_REF]. That is, if Proj Hol (M, g) contains a one parameter group of non-affine transformations, then, up to a scaling, (M, g) is holomorphically isometric to P d (C) endowed with its standard metric (where d = dim C M).

Like in the Riemannian case, we are able here to handle the discrete part of Proj Hol :

Theorem (Rigidity of h-projective transformation groups). Let (M d , g) be a compact Kähler manifold. If Aff Hol (M, g) has not finite index in Proj Hol (M, g), then, up to a scaling, (M, g) is holomorphically isometric to P d (C) endowed with its Fubini-Study metric g SF .

About the proof. V. Matveev and S. Rosemann proved their Kähler identity component version by showing that all the differential geometric tools developed in the (usual Riemannian) projective case, may be adapted to the h-projective one, and enjoy all the needed properties, see [START_REF] Matveev | Proof of the Yano-Obata conjecture for h-projective transformations[END_REF] for details. Thanks to this, we will not give details of proof in the h-projective case, because it goes exactly as in the (usual) projective one. Instead, we investigate the following new aspects in the Kähler case, in particular to in order to elucidate another use of the word "projective"! Projective vs projective. Recall that a complex manifold M is called projective if it is holomorphic to a (closed regular) complex submanifold of some projective space P N C). Endowed with the restriction of g SF , (M, g SF |M ) is a Kähler manifold. However, only few (other) Kähler metrics (M, g) admit holomorphic isometric embedding in a projective space (but, of course real analytic isometric embedding exist, by Nash Theorem). The dramatic example is that of an elliptic curve, that is a 2-torus with a complex structure. It admits a large space of holomorphic embedding in projective spaces of different dimensions, but the induced metric on them can never be flat! This is one case of a "Theorema Egregium" due to Calabi [START_REF] Calabi | Isometric imbedding of complex manifolds[END_REF] which says that holomorphic isometric immersions in space forms of constant holomorphic sectional curvature, are absolutely rigid (see §8).

Theorem 1.5. Let (M d , g) be a complex submanifold of a projective space P N (C) endowed with the induced metric (from the normalized Fubini-Study). Then the group Proj Hol (M, g) of holomorphic projective transformations is a finite extension of Iso Hol (M, g), its group of holomorphic isometries, unless (M, g) is holomorphically homothetic to P d (C). More precisely, up to composition with SU(1+N), M is the image of a Veronese map: v k : P d (C) → P N (C) (which expands the metric by a factor k).

Remark 1.6. There are submanifolds M ⊂ P N (C) with a big "projective" group, say such that G M = {g ∈ GL N+1 (C), g.M = M} is non-compact and acts transitively on M. So, G M does not act projectively with respect to the induced metric, unless M is a Veronese submanifold. The G M -action preserves another kind of geometric structures? It is however remarkable that all the automorphism group of any Kähler manifold preserves a huge class of minimal submanifolds (in the sense of Riemannian geometry), namely, complex submanifolds! 1.3. Towards the general (indefinite) pseudo-Riemannian case. It was proved in [START_REF] Matveev | Pierre Gallot-Tanno theorem for closed incomplete pseudo-Riemannian manifolds and applications[END_REF] that the quotient space Proj 0 (M, g)/Aff 0 (M, g) has always dimension ≤ 1. We have the following generalization to full groups.

Theorem 1.7. Let (M, g) be a compact pseudo-Riemannian manifold having an essential projective group, that is, Proj(M, g)/Aff(M, g) is infinite. Then, up to finite index: 1) Aff(M, g) = Iso(M, g) and it is a normal subgroup of Proj(M, g).

2) Proj(M, g)/Iso(M, g) is isomorphic to a subgroup of R. More precisely, there is a representation Proj(M, g) → SL 2 (R) whose kernel is Aff(M, g) and range contained in a non-elliptic 1-parameter group.

1.3.1. Organization. We restrict ourselves here to compact manifolds, and from §4 to the case of metrics of projective mobility dim P (M, g) = 2.

Our proofs are mostly algebraic, somewhere dynamical but rarely geometrical! Acknowledgements. I would like to thank L. Florit, A. J. Di Scala, V. Matveev and P. Mounoud for their help, and especially the referee for many interesting remarks and suggestions.

1.3.2.

Added to last version. Around one year and half after the publication of our present work (in ArXiv), V. Matveev [START_REF] Matveev | On the number of nontrivial projective transformations of closed manifolds[END_REF] improved estimate in Theorem 1.3 of the (finite) index of Aff(M, g) in Proj(M, g). He shows that this index is ≤ 2 (of course when (M, g) is not homothetic to a quotient of the standard sphere). Matveev's proof consists in pursuing analysis of our elliptic case §5.0.2 (otherwise, he bases on our results here, especially in the hyperbolic case).

ACTIONS, GENERAL CONSIDERATIONS

M is here a compact smooth manifold.

2.0.1. Let E be the space of (1, 1)-tensors T , i.e. sections of the linear bundle End(T M) → M: for any x, T x is linear map T x M → T x M. The space E has a natural structure of algebra with unit element I the identity of T M (over R as well as over C ∞ (M) or C k (M)).

Diff(M) acts naturally on E by

( f , T ) → ρ E ( f )T = f * T defined naturally by ( f * P) x = D f -1 x f T f -1 x D x f -1 .
2.0.2. Let G be the space of pseudo-Riemannian metrics on M. Then, Diff(M) acts on G by

taking direct image, ( f , g) → ρ G ( f )g = f * g defined by ( f * g) x (u, v) = g f -1 (x) ((D x f ) -1 u, (D x f ) -1 v).
2.0.3. Notation. We will sometimes use the usual notations f * T and f * g for ρ E ( f )T and ρ G ( f )g, respectively.

2.0.4. Transfer. Given a metric g 0 on M, any other metric g can by written g(., .) = g 0 (T., .), where the transfer tensor T = T (g, g 0 ) is a g 0 -symmetric(1, 1)-tensor (i.e. T x is a symmetric endomorphism of (T x M, g 0x )).

In fact, a metric g defines a bundle isomorphism T M → T * M, and thus T (g, g 0 ) = g -1 0 g.

In other words, we have a map

C g 0 : g ∈ G → T (g, g 0 ) = g -1 0 g ∈ E.
In particular C g 0 (g 0 ) = I (the identity of T M).

2.0.5. Transfer action. The transfer of the natural Diff-action ρ E on G to E by means of C g 0 , is by definition

( f , T ) → ρ GE ( f )(T ) = C g 0 (ρ G ( f )(C -1 g 0 T )) It equals: g 0 -1 ρ G ( f )(g 0 T ) = g -1 0 (ρ G ( f )g 0 )(ρ E ( f )T ) = S f ρ E ( f )T where the g 0 -strength of f is S f = g -1 0 (ρ G ( f )g 0 ).
2.0.6. A preserved functional. The following "norm-like" functional

Q(T ) = M |detT |dv g 0 is preserved by ρ GE . Indeed, Q(ρ GE ( f )T ) = M |(detS f )det( f * T )|dv g 0 = M |detT |( f -1 (x))Jac x f -1 dv g 0 ,
and this equals Q(T ).

2.0.7. Consider now the partially defined transform

F : L ∈ E → T = L -1 detL ∈ E. Its inverse map is given by F -1 (T ) = (detT ) 1 1+d T -1 (d = dim M).
It is remarkable that F commutes with the Diff-action ρ E on E. The finite dimensional version of this for a linear space

E is that u → End(E) → u -1 detu ∈ End(E) commutes with the GL(E) action given by (A, u) ∈ GL(E) × End(E) → AuA -1 ∈ End(E) 2.0.8. Action in the L-representation. Consider now the map g ∈ G → L = F -1 (C g 0 (g)) ∈ E
In other words, to a metric g, we associate the (1, 1)-tensor L such that g(., .) = 1 detL g 0 (L -1 ., .).

The corresponding action ρ on E is given by:

ρ( f )L = (ρ E ( f )L)K f where K f , the g 0 -strength of f in the L-representation, is the F -1 -transform of S f , that is K f is defined by ρ G ( f )g 0 (., .) = 1 detK f g 0 (K -1 f ., .
). Corresponding to Q, ρ preserves the partially defined functional: L → N(L) = M 1 detL (1+d)/2 dv g 0 2.0.9. The chain rule for strength.

K f n = ( f n-1 * K f )( f n-2 * K f ) . . . ( f * K f )K f (of course ( f k ) * = ( f * ) k ).
2.0.10. Summarizing: Fact 2.1. Let g 0 be a fixed metric on M. To any metric g, let L be the (1, 1) tensor defined by g(., .) = 1 detL g 0 (L -1 ., .). The Diff-action on (1, 1)tensors deduced from the usual action on metrics by means of this map g → L is given by

( f , L) ∈ Diff(M) × E → ρ( f )L = ( f * L)K f
Here K f is the L-tensor associated to f * g 0 , i.e. f * g 0 = 1 detK f g 0 (K -1 f ., .), and f * L denotes the usual action on E.

-f is an isometry of g

0 ⇐⇒ K f = I -f is a g 0 -similarity (that is f * g 0 = bg 0 for some constant b) ⇐⇒ K f = bI for some b. -ρ preserves the function L → N(L) = M 1 detL (1+d)/2 dv g 0 3. LINEARIZATION, REPRESENTATION OF Proj(M, g) IN L(M,g)
(We will henceforth mostly deal with only one metric and so we will denote it g instead of g 0 ).

3.0.1. The space L(M,g). Recall that P (M, g) denotes the class of metrics projectively equivalent to g.

Let L * (M, g) be the image of P (M, g) under the correspondence of Fact 2.1, and L(M,g) its linear hull:

L(M,g) = {L = Σ i a i L i , a i ∈ R, such that 1 detL i g(L -1 i ., .
) is projectively equivalent to g}

Let us call L-tensors the elements of this space.

3.0.2. Linearization.

Theorem 3.1. [START_REF] Bolsinov | Geometrical interpretation of Benenti systems[END_REF][START_REF] Sinjukov | Geodesic mappings of Riemannian spaces[END_REF] L ∈ L(M,g) iff L satisfies the linear equation:

g((∇ u L)v, w) = 1 2 g(v, u)dtrace(L)(w) + 1 2 g(w, u)dtrace(L)(v)
where ∇ is the Levi-Civita connection of g. Furthermore:

-

L * (M, g) is an open subset of L(M,g): an element L ∈ L(M,g) belongs to L * (M, g) iff
it is an isomorphism of T M.

-L(M,g) has finite dimension (bounded by that corresponding to the projective space of same dimension).

-L ∈ L * (M, g) is parallel iff the corresponding metric 1 detL g(L -1 ., .) is affinely equivalent to g, iff L has constant eigenvalues.

Linear representation of Proj(M, g).

Fact 3.2. We have a finite dimensional representation,

f ∈ Proj(M, g) → ρ(f) ∈ GL(L(M, g)) where ρ( f )(L) = f * (L).K f .
• ρ preserves the norm-like function N(L) = M 1 detL (1+d)/2 dv g . • Let p : GL(L(M, g)) → PGL(L(M, g)) be the canonical projection , then p is injective on ρ(Proj(M, g)), or has at most a kernel ∼ = Z/2Z.

• Let D be the subset of degenerate tensors in L(M,g):

D = {L ∈ L(M,g), L not an isomorphism of T M} Then D is a closed cone invariant under ρ.
Proof.

-The first point is imported from Fact 2.1 -For the second one, let aA and A in GL(L(M, g)) such that both preserve N, then

N(aA(L)) = N(A(L)) = N(L), for any L. But N(aL) = |a| s N(L) with s = -d(d + 1)/2,
and hence a = ±1.

-To prove ρ-invariance of D, observe that L ∈ D iff for some x ∈ M, detL(x) = 0. But

ρ( f )L = f * LK f , and hence det(ρ( f )L)( f (x)) = detL(x)detK f (x) = 0 Remark 3.3.
Actually D coincides essentially with the ∞-level of N.

4. THE CASE dim L(M,g) = 2, A HOMOGRAPHY 4.0.1. Hypothesis. Henceforth, we will assume that dim L(M,g) = 2. Fix f that is not homothetic, i.e. K = K f is not a multiple of I. Hence L(M,g) is spanned by K and I. If the spectrum is real and described by d continuous eigenfunctions x → λ 1 (x) ≤ . . . ≤

λ d (x) (d = dim M), then D = ∪ i=d i=1 (C i ∪ -C i ), where C i = {a(K -tI), a ∈ R + , and inf λ i ≤ t ≤ sup λ i } Each C i is a proper convex cone (sector).
Finally, unifying intersecting sectors, we get a minimal union:

D = ∪ i=k i=1 (D i ∪ -D i ),
where the D i are disjoint sectors.

Proof. I (as well as K) do not belong to D and hence any element of this set has the form a(K -tI). This belongs to D iff det(K(x) -tI) = 0, for some x, that is t ∈ ∪ i (Image(λ i )), and the cones C i follow.

Action by homography.

4.2.1. Equation. By the 2-dimensional assumption, there exist α, β such that:

ρ( f )(K) = ( f * K)K = αK + βI Equivalently, f * K = αI + βK -1 Say somehow formally, f * K = αK+βI K . Since f * I = I, in the basis {K, I}, ρ( f ) : L → f * (L).K f has a matrix B = B f = α 1 β 0 4.2.2. The group GL 2 (R) (more faithfully PGL 2 (R)) acts on the (projective) circle S 1 = R = R ∪ ∞, by means of the law z → A z = az + b cz + d for A = a b c d ∈ GL 2 (R)
In fact, we can also let GL 2 (R) act on the space of (1, 1)-tensors by the same formula:

(A X )(x) = (aX (x) + bI)(cX (x) + dI) -1 .
In other words, the action is fiberwise, and when a fiber 4.2.3. Iteration. We have:

End(T x M) is identified to Mat n (R), then A X = aX+b cX+d Now, the previous equation f * K = αK+βI
f * n K = A n K, for any n ∈ Z (4.1)
This can be proved in a formal way. Let C be an endomorphism on an abstract algebra

{1, x, x -1 , . . .}, such that C(x) = α + βx -1 (with C(1) = 1 and C(x -1 ) = C(x) -1
). Then,

C n (x) = A n x, where A = α β 1 0 .
4.2.4. Significance for eigen-functions. Let x ∈ M and y = f (x) and denote T = D x f :

T x M → T y M. The relation f * K = αI + βK -1 means that T -1 K y T = α + βK -1
x . This implies in particular that T maps an invariant subspace of (T x M, K x ) to an invariant subspace of (T y M, K y ). If

E λ (x) ⊂ T x M is the (generalized) K x -eigenspace associated to λ, then T maps it to E A -1 λ (y) ⊂ T y M.
Let x → Sp(x) ⊂ C be the multivalued spectrum function of K, that is Sp(x) ⊂ C is the set of eigenvalues of K x . Then the image A Sp( f (x)) (of the subset Sp( f (x)) under the homography A -1 ) equals Sp(x), and so

Sp( f (x)) = A -1 Sp(x) Also, if λ : M → C is a continuous K-eigen-function, that is λ(x) ∈ Sp(x) for any x ∈ M and λ is continuous, then x → λ ′ (x) = A -1 λ( f -1 (x))
is another continuous K-eigen-function. (3) Hyperbolic: A has two fixed points F - A and

F + A . Up to conjugacy, F - A = 0, F + A = ∞, and f (x) = ax, with 0 < a < 1. Now, if A ∈ GL + 2 (R)
, its homographic action coincides with that of A detA , and the same classification applies.

RIEMANNIAN METRICS: NON-HYPERBOLIC CASES

(M, g) is here a compact Riemannian manifold, and f as in the previous section a chosen element such that {K f , I} generate L(M,g), and f is not affine.

In this Riemannian setting, all elements of L(M,g) are diagonalizable (since self-adjoint).

Let G = ρ(Proj(M, g)), and

G + = G ∩ GL + 2 (R).
Let A = ρ(h) ∈ G + be non-trivial, then K h is not collinear to I. Indeed, assume by contradiction that K h = aI, then recall that h * g(., .) = g(S h ., .) and S h = K -1 h detK h . It follows that S h has the form S h = bI, but equality of volumes of (M, g) and (M, h * g) implies b = 1 and hence also K h = I (that is h ∈ Iso(M, g)). Thus, ρ(h)I = I.

On the other hand, by Fact 4.1, ρ(h) preserves a finite set of lines, all different from RI. Let l + 1 and l + 2 be the two nearest half lines to R + I. If ρ(h) = Id, then necessarily ρ(h)l + 1 = l + 2 , but this implies ρ(h) is a reflection which contradicts our hypothesis detρ(h) > 0.

5.0.1. G + can not contain parabolic elements. Assume by contradiction that ρ(h) is parabolic with fixed point F h . Then, F h is the unique real spectral value of K h (because there is no other bounded set of R invariant under the associated homography), and thus K h is proportional to I (since it is diagonalizable), which we have just proved to be impossible.

5.0.2.

Case where all elements of G + are elliptic. Recall that we have a union of k ≤ dim M 2) and hence cyclic. Now, if a rotation preserves a set of k-disjoint sectors, then it has order ≤ 2k. We will observe in our case that this order is in fact ≤ k.

disjoints sectors D i , such that D = ∪ i=k i=1 (D i ∪ -D i ) is G-invariant (Fact 4.1). If k > 1, then the stabilizer of D in SL 2 (R) is compact, and we can assume G is a subgroup of O(2). G + (= G ∩ GL 2 (R)) is a finite subgroup of SO(
However, in our case, we know that any ρ(h)

∈ G + is = -Id (since otherwise -I = ρ(h)I = h * (I)K h = K h
, and hence K h = -I which we have already excluded). Say, in other words we can see the rotation acting on the projective space rather than the circle and get exactly k-sectors and deduce that actually, G + has order ≤ k.

As for G (if strictly bigger than G + ), it is dihedral of order ≤ 2k.

Finally, in the case k = 1, that is D = D 1 ∪ -D 1 , its stabilizer in SL 2 (R) contains -Id together with a one parameter hyperbolic group. So, if we assume all elements of G + elliptic, we get G + = {1}. In this case, G itself reduces to a single reflection (if non-trivial).

5.0.3. About Iso(M, g). Observe first that if h ∈ Aff(M, g), then necessarily K h is proportional to I since otherwise K h will be a combination with constant coefficients of I and K h , and thus has constant eigenvalues, and therefore f ∈ Aff(M, g) (see 3.1) contradicting our hypothesis. As observed previously K h = I, that is h ∈ Iso(M, g) and so Aff(M, g) = Iso(M, g).

On the other hand, if h ∈ Iso(M, g), and ρ(h

) ∈ G + , then ρ(h) = Id. In general, if ρ(h) = Id, then it is a reflection since ρ(h 2 ) = Id.
Let Iso (2) (M, g) be the normal subgroup of Iso(M, g) generated by squares h 2 , h ∈ Iso(M, g). Then:

-either ker ρ = Iso(M, g), -or ker ρ = Iso (2) (M, g), and this has index 2 in Iso(M, g).

-in all cases, Iso(M, g) or Iso (2) (M, g) is normal in Proj(M, g), and the corresponding quotient is cyclic of order ≤ dim M, or dihedral of order ≤ 2 dim M.

RIEMANNIAN METRICS, HYPERBOLIC CASE

In the present section, (M, g) is a compact Riemannian manifold with dim L(M,g

) = 2 and f ∈ Proj(M, g) is such that ρ( f ) is hyperbolic.
The final goal (of the section) is to prove that (M, g) is projectively flat. This will be done by proving the vanishing of its Weyl projective tensor W (recalled below). For this, one iterates a vector z = W (u, v)w by the D f -dynamics to get a sequence z n = D f n z = W (D f n u, D f n v)D f n w, and shows that it has two different growth rates (when n → ±∞) unless z = 0. 6.1. Size of the spectrum. Fact 6.1. The homography A defined by ρ( f ) has two real finite fixed points λ -< λ + . K = K f has exactly one non-constant eigen-function λ. It has multiplicity 1 (at generic points), range the interval [λ -, λ + ], and satisfies the equivariance:

λ( f (x)) = A -1 λ(x).
The full spectrum of K may be {λ -, λ, λ + }, {λ -, λ} or {λ, λ + }. We denote the multiplicities of λ -and λ + by d -and d + respectively, and hence dim M = 1 + d 1 + d + .

Proof. Let µ 1 (x) ≤ . . . ≤ µ d (x) be the eigenfunctions (with multiplicity) of K(x). From 4.2.4, the map µ ′ i :

x → A -1 µ i ( f -1 (x)
) is another eigenfunction and hence equals some µ j . Taking a power of f , we can assume

µ ′ i = µ i , that is µ i ( f -1 (x)) = A µ i (x)
. In other words, µ i is an equivariant map between the two systems (M, f ) and (R, A -1 ). Thus, Image(µ i ) is a bounded A -1 -invariant interval. Hence λ ± belong to R (rather than R) and the image of µ i can be {λ -}, {λ + } or [λ -, λ + ]. The fact that only one µ i has range [λ -, λ + ] follows from the following nice fact: : Theorem 6.2. [START_REF] Matveev | Trajectory equivalence and corresponding integrals[END_REF] Let (M, g) be a complete Riemannian manifold and L ∈ L(M,g 0 ). Then two eigen-functions µ i ≤ µ j satisfy sup µ i ≤ inf µ j (that is not only µ i (x) ≤ µ j (x), but even µ i (x) ≤ µ j (y) for any x, y ∈ M).

Dynamics of f .

Define the singular sets S ± = {x ∈ M, λ(x) = λ ± }. 6.2.1. Lyapunov splitting. On M \ (S -∪ S + ), corresponding to the eigenspace decomposition of K = K f , we have a regular and orthogonal splitting

T M = E -⊕ E + ⊕ E λ .
Due to the relation, f * K = αI + βK -1 , f preserves this splitting.

Remark 6.3. Even in the linear situation of a matrix A ∈ GL d (R), it is rare that A * A and its conjugate A -1 (A * A)A have the same eigenspace decomposition! 6.2.2. Distortion. Recall the definition of the L-strength f * g(., .) = 1 detK g(K -1 ., .) vs the ordinary strength S = K -1 detK . If y = f (x) and u ∈ T y M belongs to a µ = µ S (y)-eigenspace of S, then g x (D y f -1 u, D y f -1 u) = µg y (u, u). In our case, D x f sends S-eigenspaces at x to S-eigenspaces at y, by applying a similarity of ratio 1 √ µ S (y)

.

In order to compute this by means of K-eigenvalues, observe that

detK(x) = λ d - -λ d + + λ(x) (where d -, d + are the respective dimension of E -and E + ).
Thus, for any x, D x f maps similarly E

-(x) to E -( f (x)) with similarity ratio ζ -(x) such that ζ 2 -(x) = (detK( f (x))λ -= (λ d - -λ d + + λ( f (x))λ - As for D x f : E + (x) → E + ( f (x)) and D x f : E λ (x) → E λ ( f (x))
, their respective distortions are:

ζ 2 + (x) = (λ d - -λ d + + λ( f (x)))λ + and ζ 2 λ (x) = (λ d - -λ d + + λ( f (x)))λ( f (x)) 6.2.3. Data for f -1 . Let λ * 1 , λ * -, λ * + , ζ * λ , .
. . be the analogous quantities corresponding to f -1 . Observe that f -1 preserves the same Lyapunov splitting and thus

ζ * -(x)ζ -( f -1 (x)) = 1. It follows that λ * 1 (x) = 1 λ 1 ( f -1 (x)) , λ * -= 1 λ + , and λ * + = 1 λ -.
6.2.4. Estimation of the Jacobian. From above we infer that: 1+d) , when n → +∞ (recall that

(Jac f x ) 2 = (detD x f) 2 = (ζ -(x)ζ λ (x)ζ + (x)) 2 = (λ d - -λ d + + λ(f(x))) 1+d Now, in general, Jac f n x = Jac f f n-1 x . . . Jac f x , and hence (Jac f n x ) 2 = ((λ d - -λ d + + ) n (λ(f n (x)) . . . λ(f(x)))) 1+d Fact 6.4. Assume that A -1 is decreasing on [λ -, λ + ], that is λ + is repulsing and λ -is attracting, equivalently, λ 1 is decreasing along f -orbits: λ 1 ( f (x) ≤ λ 1 (x). Then, on compact sets M \S + , (Jac f n x ) 2 is uniformly equivalent to (λ d -+1 - λ d + + ) n(
S + = {x ∈ M/λ 1 (x) = λ + }).
The proof bases on the relation λ( f k (x)) = (A -1 ) k λ(x) and the next lemma. Proof. In a small neighbourhood of λ -, the C -action is equivalent to a linear contraction fixing λ -, h : z → α(zλ -) + λ -, with 0 < α < 1. This equivalence is valid also on any compact interval

[λ -, λ + -ε]. Thus h n z = α n (z -λ -) + λ -. The above product is (cα n + 1)(cα n-1 + 1) . . . (cα + 1)(c + 1)
, where c = z-λ - λ -. This product is convergent since it can be bounded by Π i=n i=0 (e |c|α i ) ≤ e |c|(Σα i ) .

Corollary 6.6. Keep the assumption A -1 decreasing. Then (λ

d - -λ d + + )λ -≤ 1 and (λ d - -λ d + + )λ + ≥ 1. (In particular λ -< 1 < λ + ).
Proof. By the fact above, if λ

d -+1 - λ d + + > 1, then M\S -Jac f n x → ∞ when n → +∞ contradict- ing that M has a finite volume.
The other inequality holds by applying the previous fact to f -1 . Observe for this, that indeed the eigen-function λ * 1 corresponding to f -1 verifies the same decreasing hypothesis.

W : T M × T M × T M → T M, that is invariant under Proj(M, g): W (D x f u, D x f v, D x f w) = D x f (W (u, v, w)
), for any u, v, w ∈ T M (and any f ∈ Proj(M, g)) . In dimension ≥ 3, its vanishing is the obstruction to projective flatness of (M, g), that is the fact that (M, g) has a constant sectional curvature. Unlike the conformal case, the projective Weyl tensor is not a curvature tensor, that is it does not satisfy all the usual symmetries of curvature tensors (see for instance [START_REF] Besse | Einstein manifolds. Classics in Mathematics[END_REF][START_REF] Eastwood | Notes on projective differential geometry, Symmetries and Overdetermined Systems of Partial Differential Equations[END_REF] for more information). Its true definition is as follows. If u, v, w, z are four vectors in T x M such that any two of them are either equal or orthogonal (that is they are part of an orthonormal basis), then:

g x (W (u, v, w), z) = g x (R(u, v)w, z) - 1 n -1 (δ z v Ric(w, u) -δ z u Ric(w, v)) (6.1)
where Ric is the Ricci tensor and δ is the Kronecker symbol.

6.3.1. Boundedness. By compactness, W is bounded by means of g, that is W (u, v, w) ≤ C u v w , for some constant C, where . is the norm associated to g.

Notation ≈. We will deal with sequences of positive functions A(n, x) and numbers a n . We will say they have the same growth rate for x in a compact set K ⊂ M, and write A(n, x) ≈ a n , if the ratio A(n,x) a n belongs to an interval [α, β], 0 < α ≤ β < ∞, when x ∈ K. In general, K is given by the context and hence omitted. 6.3.2. Asymptotic growth under D f . The goal in this paragraph is to estimate the asymptotic behaviour under the tangent dynamics D f of vectors in each of of the Lyapunov spaces ( §6.2.1)

Case of E -: D x f n maps similarly E -(x) to E -( f n (x)) with a contraction factor ζ -(n, x) = ζ -(x)ζ -( f (x)) . . . ζ -( f n-1 (x)) More concretely, If u ∈ E -(x), then D x f n u = ζ -(n, x) u Recall that ζ -(x) = ((λ d - -λ d + + λ( f (x))λ -) 1/2 . If x ∈ M \ S + , i.e. λ(x) < λ + , then, by Lemma 6.5, ζ -(n, x) grows as ζ -(n, x) ≈ (λ d -+2 - λ d + + ) n/2 , n → +∞
Case of E + and E λ : On defines in the same way ζ + (n, x) and ζ λ (n, x) as distortion factors of

D x f n form E + (x) to E + ( f n (x)) and from E λ (x) to E λ ( f n (x))), respectively. One gets when n → ∞ ζ + (n, x) ≈ (λ d -+1 - λ d + +1 + ) n/2 ζ λ (n, x) ≈ (λ d -+2 - λ d + + ) n/2 ≈ ζ -(n, x)
In sum, when n → +∞,

If u ∈ E -(x), D x f n u = ζ -(n, x) u ≈ (λ d -+2 - λ d + + ) n/2 u If u ∈ E λ (x), D x f n u = ζ λ (n, x) u ≈ (λ d -+2 - λ d + + ) n/2 u If u ∈ E + (x), D x f n u = ζ + (n, x) u ≈ (λ d -+1 - λ d + +1 + ) n/2 u
Observe that all these behaviours are uniform on compact sets of M \ S + . Also, observe

M \ (S -∪ S + ), W (u, v, w) = 0 once all u, v, w belong to E -(x) ⊕ E λ (x), or all belong to E + x) ⊕ E λ (x).
Proof. To begin with, assume u, v, w ∈ E -(x), then and denote z n = W (D x f n u, D x f n v, D x f n w). By the two previous paragraphs,

z n ≤ C D x f n u D x f n v D x f n w = ζ -(n, x) 3 u v w ≈ (λ d -+2 - λ d + + ) 3n/2 C u v w That is, z n (λ d -+2 - λ d + + ) 3n/2
(where means that the ratio of the left hand by the right one is bounded independently of x).

On the other hand, by f -invariance of W ,

z n = W (D x f n u, D x f n v, D x f n w) = D x f n z, for z = W (u, v, w). Decompose z = z -+ z + + z λ ∈ E -(x) ⊕ E + (x) ⊕ E λ (x)
, and let z n = z - n + z + n + z λ n accordingly, i.e. z - n = D x f n z -... Thus, the norm of each of these parts is dominated by (λ

d -+2 - λ d + + ) 3n/2 . However, by §6.3.2 z - n ≈ (λ d -+2 - λ d + + ) n/2 z -. If c denotes (λ d -+2 - λ d + + )
1/2 , then recall it is < 1. So, we have at the same time z - n c 3n and z - n ≈ c n z -, which obviously implies z -= 0. Vanishing of z + and z λ is easier to check. Indeed, they behave like ζ + (n, x) z + (and ζ λ (n, x) z λ respectively). By §6.3.2, these are at least equivalent to c n z + (and c n z + respectively). They can not be dominated by c 3n , unless z + = z λ = 0.

Finally, the case where all u, v and w belong to E -(x) ⊕ E λ (x), or all belong to E + x) ⊕ E λ (x). can be handled in the same way. 6.3.4. Commutation. It was observed in particular in [START_REF] Kiosak | Proof of projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two[END_REF] that any L ∈ L(M,g) commutes with the Ricci curvature Ric (seen as an endomorphism of T M). From it we deduce that Ric(u, v) = 0 when u and v belong to two different eigenspaces of L (two commuting symmetric endomorphisms of a Euclidean space have a common eigen-decomposition, which is End of proof in higher dimension. Vanishing of the Weyl tensor in dimension ≥ 3 means that (M, g) has constant sectional curvature (see for instance [START_REF] Besse | Einstein manifolds. Classics in Mathematics[END_REF][START_REF] Eastwood | Notes on projective differential geometry, Symmetries and Overdetermined Systems of Partial Differential Equations[END_REF]). The universal cover of M is necessarily the sphere since the Euclidean and hyperbolic spaces have no projective non-affine transformations.

Proof in dimension 2. The remaining part of the present section is devoted to the case dim M = 2 and ρ( f ) hyperbolic. Our goal in the sequel is to show that in this case, too, (M, g) has constant curvature.

The spectrum of K f consists of λ, and one constant, say λ -.

7.0.1. Warped product structure. Let F -and F λ the two one dimensional foliations tangent to the eigenspaces E -and E λ . They are regular foliations on

M \ S -(recall that S -= {x/λ(x) = λ -}).
Recall that Dini normal form says that two projectively equivalent metrics g and ḡ on a surface have the following form :

g = (X (x) -Y (y)(dx 2 + dy 2 ), ḡ = ( 1
Y (y) - 1 X (x) )( dx 2 X (x) + dy 2 Y (y) )
in some coordinates system (x, y) and near any point where the (1, 1)-tensor L defined by ḡ(., .) = g( L -1 detL ., .), has simple eigenvalues. In fact, X (x) and Y (y) are the eigenvalues of L(x, y), and the coordinates are adapted to eigenspaces.

From this normal form one deduces that (F λ , F -) determines a warped product (for g as well as ḡ), that is, there are coordinates (r, θ), where 1 ∂r (resp. 1 ∂θ ) is tangent to F λ (resp. F -), and the metric has the form dr 2 + δ(r)dθ 2 (see [START_REF] Zeghib | Geometry of warped products[END_REF] for more information on warped products). Indeed here Y (y) = λ -, and hence g = (X (x) -λ -)dx 2 + (X (x) -λ -)dy 2 , and then change coordinates according to dr 2 = (X (x) -λ -)dx 2 , θ = y.

We deduce in particular that the leaves of F λ are geodesic in (M, g).

7.0.2. Topology. By 6.3.2, D x f is contracting away from S + = {x/λ(x) = λ + }. Let c ∈]λ -, λ + [ and M c = {x/λ(x) ≤ c}.
It is a codimension 0 compact submanifold with boundary the level λ -1 (c), for c generic. In particular, it has a finite number of connected components. Since λ is decreasing with f ( §6.2), f preserves M c : f (M c ) ⊂ M c . Taking a power of f , we can assume it preserves each component of M c . On such a component, say M 0 c , f contracts the Riemannian metric, and hence also its generated distance. It follows that f has a unique fixed point x 0 ∈ M 0 c . The M 0 c 's, for c decreasing to λ -, is a decreasing family converging to x 0 . It follows that these M c 's are topological discs, their boundaries λ -1 (c) are circles surrounding x 0 , and finally x 0 is the unique point in M 0 c with λ(x) = λ -.

It is a general fact on tensors of L(M,g), an eigenfunction is constant along leaves tangent to the eigenspaces associated to the other eigen-functions. In our case, λ is constant on the F --leaves, equivalently, leaves of F -are levels of λ. So, these leaves are circles surrounding x 0 . Those of F λ are orthogonal to these circles, and hence they are nothing but the geodesics emanating from x 0 . Thus, F λ and F -have x 0 as a unique singularity in M 0 c . 7.0.3. Geometry. We infer from the above analysis that the polar coordinates around x 0 gives rise to a warped product structure, that is, the metric on these coordinates (r, θ), has the form: dr 2 + δ(r)dθ 2 (in the general case δ depends rather on (r, θ)). At x 0 , D x 0 f is a similarity with coefficient λ -.

Observe next that, from the form of the metric, rotations θ → θ + θ 0 are isometries. Composing f with a suitable rotation, we can assume f is a "pure homothety", i.e. it fixes individually each geodesic emanating form x 0 . Thus f acts only at the r-level: f (r, θ) = f (r).

Now, the idea is to construct a higher dimensional example with the same ingredients, e.g. δ and f . Precisely, consider the metric dr 2 + δ(r)dΩ 2 , where dΩ 2 is the standard metric on a sphere S N . We will show in the Lemma 7.1 below that this metric is indeed smooth

We let f act by f (r, Ω) = f (r).

One verifies that f is projective. Indeed, SO(N + 1) acts isometrically and commutes with f , and any geodesic is contained in a copy of our initial surface (since, as in the case of R N+1 , for any two points x and y, one may find a copy of the surface containing them).

This new f has the same dynamical behaviour as the former one, and one proves as in §6 that the projective Weyl tensor of this new metric vanishes and it has therefore a constant sectional curvature. The same is true for our initial surface. Lemma 7.1. Consider a metric g of the form dr 2 + δ(r)dΩ 2 , where δ is defined on an interval [0, R[, smooth on ]0, R[, and dΩ 2 is the metric of S N (thus g is defined on a ball B(0, R) \ {0} in R N+1 \ {0}). Then g extends smoothly to 0 if and only if δ(r) = ζ(r 2 ), where ζ is smooth as a function of r and ζ ′ (0) = 1. In particular g is smooth for some dimension N > 0 iff it is smooth for any.

Proof. Observe firstly that the condition on δ is equivalent to that the function η(r) = δ(r)-r 2 r 4 equals κ(r 2 ) where κ is smooth on r.

Consider the mapping

Ω : z = (z 1 , . . . , z N+1 ) ∈ R N+1 \{0} → Ω(z) = z z = (Ω 1 , . . . , Ω N+1 ) ∈ S N . So dΩ = (dΩ 1 , . . . , dΩ N+1 ) is a vectorial 1-differential form on R N+1 \ {0} and dΩ 2 = ΣdΩ 2
i is a field of quadratic forms (on R N+1 \ {0}) whose restriction to S N coincide with the induced metric.

Similarly, r = z , and hence dr = 1 r (Σz i dz i ). Thus r 2 dr 2 = (Σz i dz i ) 2 . On the other hand, it is known that g E = r 2 + r 2 dΩ 2 is the Euclidean metric Σdz 2 i . Therefore r 4 dΩ 2 is smooth and equals exactly:

r 4 dΩ 2 = r 2 (g E -dr 2 ) = (Σz 2 i )(Σdz 2 i ) -(Σz i dz i ) 2 = Σ i = j z 2 i dz 2 j -z i z j dz i dz j Now, let g = dr 2 + δ(r)dΩ 2 . Then g-g E = (dr 2 +δ(r)dΩ 2 )-(dr 2 +r 2 dΩ 2 ) = (δ(r)-r 2 )dΩ 2 = (δ(r) -r 2 ) r 4 (r 4 dΩ 2 ) = η(r)(r 4 dΩ 2 )
We deduce in particular that a sufficient condition for g to be smooth (as a function of z) is that η is smooth (as a function of z). To see that this is also a necessary condition, we infer from the previous formula for r 4 dΩ 2 that η(r)r 4 dΩ 2 is smooth iff the functions η(r)z i z j are smooth for any i, j. Then apply the next lemma: Lemma 7.2. Let η(r) a function such that all the functions η(r)z i z j are smooth on z. Then η(r) is smooth (as a function of z) and it equals κ(r 2 ) where κ is smooth as a function of r.

Proof. First, η(r)z 2 i → 0 when r → 0. Indeed if not, this limit does not depend on i, and one can take the ratio 1 = lim

η(r)z 2 i η(r)z 2 j = lim z 2 i z 2 j
, but the latter limit does not exist.

For the next step, to simplify notations, let us assume the dimension is 2 and note x = z 1 , y = z 2 (the proof in higher dimension is identical).

By hypothesis T (x, y) = η(r)(x 2 + y 2 ) = η(r)r 2 is smooth. Its Taylor expansion allows one to write it, up to any order, as a sum of homogeneous polynomials on x and y. Since T is SO(2)-invariant, the same is true for these polynomials. Now, let P such a polynomial of degree k. By homogeneity and SO(2)-invariance

P (x 2 +y 2 ) k 2
is constant on S 1 and hence constant, that is P is proportional to r k 2 . This implies in particular that k is even. Therefore, the Taylor expansion is on the powers r 2 , r 4 , r 6 , . . .. We finally get that η(r) has a Taylor expansion on 1, r 2 , r 4 , . . ...., which ensures the existence of κ.

Coming back to the proof of Lemma 7.1, assume now that g = dr 2 + δ(r)dΩ 2 is a smooth Riemannian metric on a neighbourhood of 0. For fixed Ω, the ray r → (r, Ω) is an arc-length parameterized geodesic. It follows that (r, Ω) → z = rΩ are normal coordinates, i.e. the inverse of the exponential map at 0. Thus, as in the Euclidean case, the metric g is smooth with respect to z. Therefore, δ(r) satisfies the same conditions as above.

Remark 7.3. As an alternative for all this proof in dimension 2, the referee suggests to mimic the higher dimensional proof by replacing the Weyl tensor by its 2-dimensional version, the Liouville tensor (as used in [START_REF] Bryant | A solution of a problem of Sophus Lie: normal forms of two-dimensional metrics admitting two projective vector fields[END_REF]).

8. THE KÄHLER CASE: PROOF OF THEOREM 1.5 Let F(N, b) denote the simply connected Hermitian space of dimension N and constant holomorphic sectional curvature b. Calabi proved (in his thesis) the following striking fact: Theorem 8.1 (Calabi [9]). Let M be a Kähler manifold (not necessarily complete) and f : M → F(N, b) a holomorphic isometric immersion. Then, f is rigid in the sense that any other immersion f ′ is deduced from f by composing with an element of Iso(F(N, b)) (this element is unique if the image of f in not contained in a totally geodesic proper subspace of F(N, b)). In particular, f is equivariant with respect to some faithful representation Iso(M) → Iso(F(N, b)).

As for holomorphic isometric immersions between space forms, one deduces (for more information, see for instance in [START_REF] Takeuchi | Homogeneous Kähler submanifolds in complex projective spaces[END_REF][START_REF] Hulin | Kähler-Einstein metrics and projective embeddings[END_REF][START_REF] Di Scala | Kähler Immersions of Homogeneous Kähler Manifolds into Complex Space Forms[END_REF]): Theorem 8.2. -The Kähler Euclidean space C d can not embed holomorphically isometrically in a projective space P N (C) (a radially simple example of this is the situation of a holomorphic vector field; it does not act isometrically, in particular its orbits are not metrically homogeneous).

-Up to ambient isometry, the holomorphic homothetic embeddings between projective spaces are given by Veronese maps: v k :

(P d (C), g FS ) → (P N (C), 1 k g FS ), N = d+k k -1, v k : [X 0 , . . . , X d ] → [. . . X I . . .]
, where X I ranges over all monomials of degree k in X 0 , . . . , X d .

Proof of Theorem 1.5. This will follow from our rigidity theorem of the h-projective group of Kähler manifolds (see §1.2), together with the following fact. Proof. This is a standard idea (see for instance [START_REF] Kobayashi | Transformation groups in differential geometry[END_REF][START_REF] Zeghib | Sur le groupe affine d'une variété riemannienne compacte[END_REF]), the unique special fact we use here is that, by Calabi Theorem, the universal cover has no flat factor in its De Rham decomposition. Thus M is a product M1 × . . . × Mm of irreducible Kähler manifolds. The holonomy group Hol M equals the product Hol M1 × . . . × Hol Mm . An affine transformation f commutes with Hol and hence preserves the De Rham splitting. Taking a power, we can assume that f actually preserves each factor, and we will thus prove that f is isometric. Since Mi is irreducible, f induces a homothety on it, say of distortion c. If c = 1, then f or f -1 is contracting with respect to the distance of Mi . In this case, f will have a (unique) fixed point in Mi . However, f preserves the Riemann curvature tensor R(X ,Y )Z of Mi . But being invariant by a contraction (or a dilation), this tensor must vanish, that is Mi is flat, contracting the fact that it is irreducible. Therefore, c = 1, that is f is isometric.

Remarks 8.4. 1. By equivariance, Segre maps P m (C) × P n (C) → P (m+1)(n+1)-1 (C) are homothetic. In particular, some (P 1 (C) × P 1 (C), 1 k (g FS ⊕ g FS )) can be embedded in some (P N (C), g FS ). By composing Veronese and Segre maps, one can also realize some metrics (P 1 (C), 1 k g FS ) × (P 1 (C), g FS )).

2. In fact, it turns out that for M a submanifold of P N (C), De Rham decomposition applies to M itself; that is the splitting of M descends to a one of M. I am indebted to A. J. Di Scala for giving me a proof of that using Calabi rigidity. Indeed, this rigidity has the following amuzing corollary: if M is holomorphically isometrically embedded in P N (C), then neither a cover nor a quotient of it can be embedded so. Now, the De Rham splitting of M gives an immersion into products of projective spaces. Segre map is isometric form this product to one big projective space, which gives us another holomorphic isometric immersion of M. But this must coincide with the immersion given by the universal cover M → M. This implies that the De Rham decomposition is defined on M itself.

FACTS ON THE INDEFINITE PSEUDO-RIEMANNIAN CASE: PROOF OF THEOREM 1.7

Let (M, g) be a compact pseudo-Riemannian manifold with projective degree of mobility dim L(M,g) = 2, such that Proj(M, g)/Aff(M, g) is infinite. Consider ρ : Proj(M, g) → GL 2 (R).

Denote G = ρ(Proj(M, g)). Theorem 1.7 says that, up to finite index, ker ρ = Iso(M, g) = Aff(M, g), and G lies in a non-elliptic one parameter group. 9.1. "Projective linear" action of Proj(M, g). So far, we singled out an element f ∈ Proj(M, g) and associate to it a homography A acting on R. It turns out that this A is nothing but that corresponding to the (projective) action of ρ( f ) on the projective space of L(M,g), identified to P 1 (R), via the basis {K = K f , I}.

Indeed, the choose of the basis {K, I}, say co-ordinates (k, i), allows one to identify P(L(M, g)) with P 1 (R). In the affine chart [k : i] ∈ P 1 (R) → z = k i , the projective action of ρ( f ) is z → αz+β z , where α and β are defined by ρ( f )K = αK + βI, as in §4. Now, we let the whole group Proj(M, g) act by means of ρ on the projective space, and in fact the complex one. More precisely, let

Φ : Proj(M, g) → PGL(L(M, g) ⊗ C)
be the action associated to ρ on P(L(M, g) ⊗ C), the projective space of the complexification of L(M,g).

The degeneracy set D is complexified as

D C = {L ∈ P 1 (L(M, g) ⊗ C), L not an isomorphism of T M ⊗ C}
The proof of the following fact is similar to that of Fact 4.1.

Fact 9.1. Let f be any element of Proj(M, g) with K f = -± I, then D C can be computed by means of K f as follows. Under the identification of P(L(M, g) ⊗ C) with P 1 (C) via the basis {K f , I}, the set D C corresponds to the range of the spectrum mapping of K f :

x ∈ M → Sp K f (x) = Spectre of K f (x) ⊂ C.
The point is that this set is invariant under the G-action. 9.1.1. By Fact 3.2, the projection of G on its image in PGL 2 (R) has finite index. In fact, since we are interested in objects up to finite index, for simplicity seek, we will argue as if G is contained in PGL 2 (R), in fact in SL 2 (R) to be more concrete. 9.1.2. The Kernel of ρ. Let h ∈ Aff(M, g), we will prove that ρ(h) = 1, up to index 2. Since h is affine, all K h -eigenvalues are constant. It follows that K h has the form aI, since otherwise it generates together with I the whole L(M,g), and hence all the K f will have constant eigenvalues for any f , contradicting the fact that Proj(M, g) Aff(M, g). By finiteness of the volume, a = ±1, say a = 1, i.e. K h = I. Now, ρ(h)L = h * LK h = h * L, and thus ρ(h)I = I. Therefore, if ρ(h) = 1, ρ(h) will be parabolic with unique fixed point I (in P 1 (C)). So, any closed ρ(h)-invariant set contains I. But this is not the case of the degeneracy set D C (since it corresponds to the spectrum). 9.2. Proof that G is contained in a one parameter group. As suggested by the referee, we will make use of Theorem 1.11 of [START_REF] Bolsinov | Splitting and gluing lemmas for geodesically equivalent pseudo-Riemannian metrics[END_REF]. It states that if for some x, K f (x) has a complex eigenvalue λ, then this is a constant eigenvalue, that is, λ is eigenvalue of K f (y) for any y ∈ M. So the proof will be essentially similar to the Riemannian case. More precisely, let f ∈ Proj(M, g) such that ρ( f ) is hyperbolic or parabolic, then K f has everywhere a real spectrum. Indeed, otherwise, the homography associated to f will have a non-real fixed point in P 1 (C) which is impossible (since this homography is real).

Furthermore, the range of the spectrum of K f is a compact interval in R (non-reduced to a point since f is not affine). Now, a parabolic homography preserves no non trivial compact interval, and so this case is impossible. In the hyperbolic case, the unique nontrivial invariant interval is that joining the two fixed points. It follows that D C is an interval in P 1 (R) ⊂ P 1 (C). Therefore, the group G preserves a subset of two points consisting in the extremities of this interval. But the subgroup of SL 2 (R) preserving two points in P 1 (R) has a one parameter subgroup as a normal subgroup of index two (e.g. in the case of {0, ∞}, this group is generated by of z → az, a > 0, and z → 1 z ). 9.2.1. Elliptic case. It remains now to consider the case where all the elements of G are elliptic, the goal here is to prove that G is finite. Let Ḡ be the closure of G and Ḡ0 its identity component. Thus Ḡ0 is a connected subgroup of SL 2 (R). It can not be SL 2 (R) since the set of elliptic elements there is not dense. The 4 others possibilities for nontrivial connected subgroups are, up to conjugacy: the affine group Aff(R) (upper triangular elements of SL 2 (R) or a one parameter of hyperbolic, parabolic or elliptic type. But, the set of elliptic elements is dense (actually just non-trivial) only in the case of an elliptic one parameter group. Hence, if non-trivial, Ḡ0 is conjugate to SO(2). The group G itself is contained in the normalizer of Ḡ0 which also equals SO(2). We will now find a contradiction leading to that this situation is impossible. Indeed, since G is dense in SO(2), its orbits in P 1 (R) are dense, and hence any G-invariant closed set in P 1 (R) equals P 1 (R). This implies that D C ∩ P 1 (R) = / 0, since this closed G-invariant subset that does not contain ∞. In sum, the spectrum of K f is nowhere real. As above, by [START_REF] Bolsinov | Splitting and gluing lemmas for geodesically equivalent pseudo-Riemannian metrics[END_REF], this implies K f has a constant spectrum and hence f is affine, but we have already excluded this possibility.

Let us consider now the case where Ḡ0 = 1 which means that G is discrete. Any element A ∈ G is elliptic and hence conjugate to an element of SO(2) which has a finite order (by discreetness). Apply Selberg Lemma (see for instance [START_REF] Alperin | An elementary account of Selberg's lemma[END_REF]), which says that a finitely generated subgroup of GL n (R) has a torsion free finite index subgroup (i.e. with no elements of finite order). Let G ′ be a finitely generated subgroup of G. Since all elements of G ′ have finite order, Selberg Lemma implies that G ′ is finite. However, a finite non-trivial subgroup of SL 2 (R) is conjugate to a unique one parameter elliptic subgroup (geometrically, it a has a unique fixed point in the hyperbolic plane). Say, if an element A ∈ G, up to conjugacy belongs to SO(2), then, for any B ∈ G, the group G ′ generated by A and B must be contained in SO [START_REF] Aminova | Pseudo-Riemannian manifolds with general geodesics[END_REF], and therefore G ⊂ SO(2). As above, G can not be dense in SO(2) and is hence finite.

We have thus proved that in all cases and after neglecting finite index objects, ρ(Proj(M, g)) lies in a hyperbolic or parabolic one parameter group, which completes the proof of Theorem 1.7.

Remark 9.2. In higher dimensions, i.e. for subgroups of SO(1, n), n > 2, it is not longer true that having all its elements elliptic implies the subgroup is contained in a compact subgroup, see [START_REF] Waterman | Purely elliptic Möbius groups. Holomorphic functions and moduli[END_REF] Remark 9.3. Let P the one parameter group that contains G (up to finite things). Then G may be equal to P, or dense (and = P), or finally discrete and hence cyclic generated by a single element. The case G = P means that M has a projective vector field. One may ask if the dense case may happen, that is if G is dense, then necessarily G = P?

4. 1 .

 1 The degenerate set D.

Fact 4 . 1 .

 41 The subset of degenerate tensors (defined above) satisfies:D = {a(K -tI), a ∈ R,and t a real spectral value of K : det(K(x) -tI) = 0 for some x} In particular I and K f / ∈ D.

K

  can be interpreted by that the linear f * -action on K equals the homographic action A K where A = α β 1 0 is the transpose of B.

4. 3 .

 3 Classification of elements of SL 2 (R). Recall that non trivial elements A of SL 2 (R) split into three classes: (1) Elliptic: A is conjugate in SL 2 (R) to a rotation, i.e. an element of SO(2). Its homographic action on R = R ∪ ∞, as well as on C = C ∪ ∞ is conjugate to a rotation. (2) Parabolic: A is unipotent, i.e. (A -1) is nilpotent. Its homographic action on R as well as on C is conjugate to a translation. It has a unique fixed point F A ∈ R. Up to conjugacy F A = ∞, and A z → z + a, where a ∈ R. It follows that if C ⊂ R is a bounded A -invariant set, then C = {F A } (and necessarily F A = ∞).

Lemma 6 . 5 .

 65 Let C be a hyperbolic element of SL 2 (R) with fixed points λ -< λ + , with λ - attracting. The sequence (C n z)(C n-1 z) . . . (C z) λ n converges simply in [λ -λ + [ to a continuous function. The convergence is uniform in any compact subset of [λ -, λ + [.

Fact 8 . 3 .

 83 Let (M d , g SF |M ) be a submanifold of (P N (C), g SF ), then Aff(M d , g SF |M )/Iso(M d , g SF|M ) is finite (vaguely bounded by n 2 !).

  6.2.5. Justification of the decreasing hypothesis forA -1 . Let us see what happens if A -1 was increasing in [λ -, λ + ].In this case, the volume estimate would give (λ -≥ 1, which leads to the contradiction λ + ≤ λ -. 6.3. The projective Weyl tensor. This is a (3, 1)-tensor

	and (λ	d --λ d + + )λ	d --λ	d + + )λ + ≤ 1

  that, surely, ζ -(n, x) is exponentially decreasing. Indeed, if not, λ -< λ + , ζ + (x, n) would be exponentially increasing. On the other hand, the Jacobian of D f n is at least equivalent to a power of ζ + (x, n) (since ζ -and ζ λ are bounded from below).

	λ 6.3.3. A first vanishing.	d -+2 -	λ	d + + ≥ 1, and since
	Fact 6.7. For any x ∈

Partially supported by the ANR project GR-Analysis-Geometry.

1

furthermore orthogonal). More precisely Ric(u, v) = 0 when u ∈ E λ (x) and v ∈ E ± (x), or u ∈ E -(x) and v ∈ E + (x). 6.3.5. A second vanishing. Fact 6.8. Let u ∈ E -(x) and v, w ∈ E + (x) ⊕ E λ (x), then W (u, v, w) ∈ E -(x).

.

Proof.

We have to prove that g x (W (u, v, w), z) = 0, whenever

Recall the defining formula 6.1 of the Weyl tensor. Observe that δ z u = 0 and Ric(w, u) = 0 (because of §6.3.4), and thus g x (W (u, v, w), z) = g x (R(u, v)w, z)

On the other hand

Proof. We know by above that z = W (u, v, w) belongs to E -.

By §6.3.2, we can write:

, where z n and u n belong to E -( f n (x)), and z n = z and u n = u .

Thus z n = W (u n , D x f n v, D x f n w). But D x f n v and D x f n w tend to 0 when n → -∞, hence z = z n → 0, that is z = 0.

6.3.6. Full vanishing. Similar to the above situation in Fact 6.8, we consider the case where u, v ∈ E + (x) ⊕ E λ (x) and w ∈ E -(x) and prove that W (u, v, w) ∈ E -(x). For this goal, observe that Ric(w, u) = Ric(w, v) = 0, and hence g x (W (u, v, w), z) = g x (R(u, v)w, z). On the other hand,

and thus W (u, v, z) = 0, and consequently g x (W (u, v, w), z) = 0 as claimed.

Next, the proof of Fact 6.9 applies here and yields that W (u, v, w) = 0. So, by this and the previous facts, we get that W (u, v, w) = 0 whenever (at least) two of them are in

Obviously, we can switch roles of E + and E -, and so get that W (u, v, w) = 0 in all cases.

7. RIEMANNIAN CASE, PROOF THEOREM 1.3

As previously, (M, g) is a compact riemannian manifold with Proj(M, g) Aff(M, g). By [START_REF] Matveev | Proof of the projective Lichnerowicz-Obata conjecture[END_REF], the degree of projective mobility dim L(M,g) equals 2. Pick f ∈ Proj(M, g) as in §4.

If ρ( f ) is non-hyperbolic for any choice of such f , then, by §5, Proj(M, g)/Iso(M, g) is finite as stated in Theorem 1.3.

If ρ( f ) is hyperbolic, then by §6, the projective Weyl tensor of (M, g) vanishes.