Fida El Chami
email: fchami@ul.edu.lb

Georges Habib
email: ghabib@ul.edu.lb

Ola Makhoul
email: ola.makhoul@ul.edu.lb

Roger Nakad
email: rnakad@ndu.edu.lb

Eigenvalue Estimate for the basic Laplacian on manifolds with foliated boundary

Keywords: Riemannian flow, manifolds with boundary, basic Laplacian, eigenvalue, second fundamental form, O'Neill tensor, basic Killing forms, rigidity results Mathematics Subject Classification: 53C12, 53C24, 58J50, 58J32

On a compact Riemannian manifold whose boundary is endowed with a Riemannian flow, we give a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic 1-forms. The equality case gives rise to a particular geometry of the flow and of the boundary. Namely, we prove that the flow is a local product and the boundary is η-umbilical. This allows to characterize the quotient of R×B by some group Γ as the limiting manifold. Here B denotes the unit closed ball. Finally, we deduce several rigidity results describing the product S 1 × S n as the boundary of a manifold.

Introduction

The Reilly formula on compact manifolds with smooth boundary has been used to estimate the eigenvalues of the Laplacian acting on functions. The aim is to state rigidity results that arise from the optimality of those estimates [START_REF] Choi | A first eigenvalue estimate for minimal hypersurfaces[END_REF][START_REF] Reilly | Application of the Hessian operator in a Riemannian manifold[END_REF][START_REF] Xia | Rigidity of compact manifolds with boundary and non-negative Ricci curvature[END_REF]. For example, C. Xia [START_REF] Xia | Rigidity of compact manifolds with boundary and non-negative Ricci curvature[END_REF] gave a lower bound for the first eigenvalue in terms of the lowest bound of the principal curvatures, assumed to be positive. The equality case is mainly characterized by the closed ball. As a direct application, he proved the following rigidity result (see [START_REF] Xia | Rigidity of compact manifolds with boundary and non-negative Ricci curvature[END_REF]Thm. 2]): Assume that on a compact domain N with a non-negative Ricci curvature, the boundary M is isometric to the round sphere with a non-negative mean curvature and that the sectional curvature of N vanishes along planes tangent to M, the domain is then isometric to the unit closed ball. This result answered partially a question proposed by Schroeder and Strake in [START_REF] Schroeder | Rigidity of convex domains in manifolds with nonnegative Ricci and sectional curvature[END_REF]Thm. 1].

In [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF], S. Raulot and A. Savo generalized the Reilly formula to differential p-forms on M (see Equation [START_REF] Kacim | Opérateurs transversalement élliptiques sur un feuilletage riemannien et applications[END_REF]) by integrating the Bochner formula and using the Stokes' theorem. Among the terms in the generalized Reilly formula, an extension of the second fundamental form S to differential p-forms is defined in a canonical way so that its eigenvalues, called the p-curvatures, depend mainly on the principal curvatures (see Section 2 for the definition). Therefore, with the use of the Reilly formula, they obtained a lower bound for the first eigenvalue λ 1,p of the Laplacian acting on exact p-forms in terms of those p-curvatures. In fact, they proved that on a compact Riemannian manifold (N n+2 , g) with non-negative curvature operator such that the p-curvatures of its boundary M are bounded from below by σ p (M ) > 0, we have for any p, 1 ≤ p ≤ n+2 2 , λ 1,p ≥ σ p (M )σ n-p+2 (M ).

(

The equality holds if and only if N is isometric to the Euclidean ball. In particular, they showed that when the boundary is isometric to the round sphere and the p-curvatures are bounded from below by p, this forces N to be isometric to the Euclidean unit ball. We notice that the above estimate generalized the one in [START_REF] Xia | Rigidity of compact manifolds with boundary and non-negative Ricci curvature[END_REF] since for p = 1 the first eigenvalue on exact 1-forms is also the first eigenvalue on functions.

In this paper, we will consider a different geometric situation. Assume that on a given Riemannian manifold N the boundary M is endowed with a Riemannian flow, that is a Riemannian foliation of 1-dimensional leaves (see Section 2). It is then natural to ask whether the above results can be generalized to the basic Laplacian defined on basic forms in order to deduce a foliated version of the mentioned rigidity results. Recall that basic forms are differential forms on M that are constants along the leaves of the flow. In this context, the geometry of the transverse structure will clearly play an essential role since many additional terms will be involved in the estimate such as the O'Neill tensor (see Section 2 for the definition).

We state our main estimate:

Theorem 1.1 Let (N n+2 , g) be a Riemannian manifold with non-negative curvature operator whose boundary M has a positive n-curvature σ n (M ). Assume that M is endowed with a minimal Riemannian flow given by a unit vector field ξ. Then

λ 1,1 ≥ σ n+1 (M ) σ 2 (M ) -sup M g(S(ξ), ξ) , (2)
where λ 1,1 denotes the first positive eigenvalue of the basic Laplacian restricted to basic closed 1-forms.

The key point in the proof of Theorem 1.1 is to show, under some curvature assumptions, that any basic closed p-form on M can be extended to a unique p-form which is closed and co-closed on the whole manifold (see Lemma 3.1). This can be done with the help of some boundary problem established in [START_REF] Schwarz | Hodge decomposition-A method for solving boundary value problems[END_REF]Lemma 3.4.7]. We point out here that the case p > 1 will be studied in a forthcoming paper. This is due to the difference in the estimation (more terms will be involved) and later in the equality case.

The equality case in (2) is of special interest since it implies that the flow is a local product and the boundary is η-umbilical. This means the second fundamental form vanishes in the direction of the vector field ξ and is a multiple of the identity in the orthogonal direction to the leaves (see Lemmas 4.1 and 4.2). Mainly, we get:

Theorem 1.2 Under the assumptions of Theorem 1.1 with σ 1 (M ) ≥ 0, if the equality case is realized in (2), the manifold M is then isometric to the quotient Γ \ R × S n and N is isometric to Γ \ R × B , for some fixed-point-free cocompact discrete subgroup Γ ⊂ R × SO n+1 , where B is the unit closed ball in R n+1 .
The problematic part in the last result is to prove that the extension ξ of the vector field ξ (which is then parallel) to the whole manifold, coming from the boundary problem in [START_REF] Schwarz | Hodge decomposition-A method for solving boundary value problems[END_REF], is also parallel on N . Therefore, we show that any connected integral submanifold of (R ξ) ⊥ is isometric to the unit closed ball by using the result of Raulot and Savo. At the end of Section 4, we prove through examples that, depending on the dimension n, the converse of Theorem 1.2 is not true in general.

Using the limiting case of Inequality (2), we end the paper by stating several rigidity results on manifolds with foliated boundary (see Section 5). Indeed, we consider the situation where the boundary of a compact manifold carries a solution of the so-called basic special Killing form. In particular, we prove that, depending on the sign of the lowest principal curvature, the following result holds:

Corollary 1.3 Let N be a (n + 2)-dimensional compact manifold with non-negative curvature operator. Assume that the boundary M is S 1 × S n and g(S(ξ), ξ) ≤ 0. If the inequality σ 2 (M ) ≥ 1 holds, the manifold N is isometric to S 1 × B .

The vector field ξ is in this case the unit vector field tangent to S 1 . Also, we prove the following:

Corollary 1.4
Let N be a (n + 2)-dimensional compact manifold with non-negative curvature operator. Assume that the boundary M is S 1 × S n and g(S(ξ), ξ) ≥ 0.

If the inequality σ 2 (M ) ≥ 1 + sup M g(S(ξ), ξ) holds, the manifold N is isometric to S 1 × B .
Finally, we point out that rigidity results on manifolds with foliated boundary have been studied in the context of spin geometry in [START_REF] Chami | Rigidity results for spin manifolds with foliated boundary[END_REF] which could be seen as a foliated version of the results in [START_REF] Hijazi | Extrinsic Killing spinors[END_REF][START_REF] Hijazi | Dirac operators on embedded hypersurfaces[END_REF][START_REF] Raulot | Rigidity of compact Riemannian spin manifolds with boundary[END_REF].

Preliminaries

In the first part of this section, we recall some basic ingredients on Riemannian flows defined on a Riemannian manifold. For more details, we refer to [START_REF] Carrière | Flots riemanniens, structure transverse des feuilletages[END_REF][START_REF] Ph | Foliations on Riemannian manifolds[END_REF].

Let (M n+1 , g) be a Riemannian manifold and let ξ be a smooth unit vector field defined on M . We say that ξ defines a Riemannian flow on M if the following relation holds (L ξ g)(Y, Z) = 0 [START_REF] Carrière | Flots riemanniens, structure transverse des feuilletages[END_REF][START_REF] Reinhart | Foliated manifolds with bundle-like metrics[END_REF] for any vector fields Y, Z orthogonal to ξ. In other words, the vector field ξ defines a foliation (by its integral curves) on M such that the metric is constant along the leaves. It is not difficult to check that, on a Riemannian flow, the endomorphism h := ∇ M ξ (known as the O'Neill tensor) defines a skew-symmetric tensor field on the normal bundle Q = ξ ⊥ → M . Therefore, one can associate to h a differential 2-form Ω given for all sections Y, Z ∈ Γ(Q) by Ω(Y, Z) = g(h(Y ), Z). On the other hand, by inducing the metric g from the manifold M to the normal bundle Q, one can define a covariant derivative ∇ (called transversal Levi-Civita connection) on sections of Q compatible with such a metric. Namely, it is defined for any section Y on Q by

∇ X Y =:    π[X, Y ] if X = ξ, π(∇ M X Y ) if X ⊥ ξ,
where π : T M → Q denotes the orthogonal projection (see [START_REF] Ph | Foliations on Riemannian manifolds[END_REF]). The curvature of the normal bundle (as a vector bundle) associated with the connection ∇ satisfies the property ξ R ∇ = 0. Moreover, one can easily check that the corresponding Levi-Civita connections on M and Q are related for all sections Z, W in Γ(Q) via the Gauss-type formulas:

   ∇ M Z W = ∇ Z W -g(h(Z), W )ξ, ∇ M ξ Z = ∇ ξ Z + h(Z) -κ(Z)ξ, (3)
where κ := ∇ M ξ ξ is the mean curvature of the flow. A flow is said to be minimal if its mean curvature κ is zero.

The set of basic forms Ω B (M ) is the set of all differential forms ϕ on M such that ξ ϕ = 0 and ξ d M ϕ = 0. This can be seen locally as the set of all forms on M which just depend on the transverse variables. Clearly from the definition, these forms are preserved by the exterior derivative d The operator ∆ b is an essentially self-adjoint operator and transversally elliptic and therefore it has a discrete spectrum by the spectral theory of transversal elliptic operators [START_REF] Kacimi | Stabilité du caractère kählérien transverse[END_REF][START_REF] Kacim | Opérateurs transversalement élliptiques sur un feuilletage riemannien et applications[END_REF].

In the next part, we will give a brief overview on manifolds with boundary and the Reilly formula that could be found in details in [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF].

Let (N n+2 , g) be a Riemannian manifold of dimension n + 2 with boundary M and let ν be the inward unit nomal vector field on M . The shape operator (or the Weingarten tensor) is a symmetric tensor field on T M defined for all X ∈ Γ(T M ) by S(X) = -∇ N X ν where ∇ N is the Levi-Civita connection of N . At a point x ∈ M , we denote by η 1 (x), • • • , η n+1 (x) the eigenvalues of S (that are called principal curvatures) and we arrange them in a way such that η

1 (x) ≤ η 2 (x) ≤ • • • ≤ η n+1 (x).
For any integer number p in {1, • • • , n + 1}; the p-curvatures σ p of M are defined by σ p (x) = η 1 (x)+• • •+η p (x). It is a clear fact that for any two numbers p and q such that p ≤ q, the inequality

σp(x) p ≤ σq(x) q
holds where the equality is achieved if and only if

η 1 (x) = η 2 (x) = • • • = η q (x) or p = q.
The p-curvatures are the lowest eigenvalues of the tensor S [p] , the canonical extension of S, to differential p-forms on M. Given any p-form ϕ on M , this tensor is defined as

S [p] (ϕ)(X 1 , • • • , X p ) = p i=1 ϕ(X 1 , • • • , S(X i ), • • • , X p ),
where X i are vector fields on M for i = 1, • • • , p. By convention, we set S [0] = 0. One then can easily check that the following inequality

S [p] (ϕ), ϕ ≥ σ p (M )|ϕ| 2 (4)
holds, where σ p (M ) is the infimum over M of the p-curvatures σ p . Moreover, we have the following property S

[p+1] (X ∧ ϕ) = S(X) ∧ ϕ + X ∧ S [p] (ϕ), (5)
for all X ∈ Γ(T M ). Indeed, for any vector fields X 1 , . . . , X p+1 , we write

S [p+1] (X ∧ ϕ)(X 1 , . . . , X p+1 ) = p+1 j=1 (X ∧ ϕ)(X 1 , . . . , S(X j ), . . . , X p+1 ) = p+1 i=1 (-1) i+1 g(X, S(X i ))ϕ(X 1 , . . . , Xi , . . . , X p+1 ) + p+1 i,j=1 i =j (-1) i+1 g(X, X i )ϕ(X 1 , . . . , Xi , . . . , S(X j ), . . . , X p+1 ) = (S(X) ∧ ϕ)(X 1 , . . . , X p+1 ) + (X ∧ S [p] (ϕ))(X 1 , . . . , X p+1 ),
where we identify vector fields with the corresponding 1-forms.

We finish this section by stating the Reilly formula established in [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF]Thm. 3]. As we already mentioned in the introduction, this formula comes from the integration of the Bochner-Weitzenböck formula over N (assumed to be compact) of the Hodge Laplacian and the use of the Stokes' formula.

More precisely, we let J * the restriction of differential forms on N to the boundary M . For any p-form α on N , the Reilly formula is then

N |d N α| 2 v g + N |δ N α| 2 v g = N |∇ N α| 2 v g + N W [p] N (α), α v g +2 M ν α, δ M (J * α) v g + M B(α, α)v g , (6)
where v g is the volume element of g on N and on its boundary M ,

B(α, α) = S [p] (J * α), J * α + S [n+2-p] (J * ( * N α)), J * ( * N α) = S [p] (J * α), J * α + (n + 1)H|ν α| 2 -S [p-1] (ν α), ν α ,
and W

[p] N = n+2 i,j=1 e j ∧ e i R(e i , e j ) is the curvature term in the Bochner formula with {e i } i=1,••• ,n+2 is a local orthonormal frame of T N and " * N " is the Hodge star operator on N . One can prove that J * ( * N α) is equal (up to a sign) to * M (ν α) where " * M " denotes the Hodge star operator on M. We also have |J * α| 2 + |ν α| 2 = |α| 2 at any point x ∈ M. We point out that if the curvature operator of N is non-negative, the term W

[p]

N is also non-negative for all p (see [START_REF] Gallot | Sur la première valeur propre du p-spectre pour les variétés à opérateur de courbure positif[END_REF]). In the last part of this section (see [START_REF] Schwarz | Hodge decomposition-A method for solving boundary value problems[END_REF] for more details), we define the space of harmonic fields H p (N ) as being the set of closed and co-closed p-forms on N. When the manifold N is compact, the space H p (N ) decomposes as [22, Thm. 2.4.8]

H p (N ) = H p D (N ) ⊕ H p co (N ),
where

H p D (N ) := {α ∈ H p (N )| J * α = 0} is the Dirichlet harmonic space and H p co (N ) := {α ∈ H p (N )| α = δ N β}
is the space of co-exact harmonic field. This splitting, known as the first Friedrich's decomposition, is L 2 -orthogonal. Finally, we recall that given any p-form ϕ on M , the boundary problem

   ∆ N φ = 0 in N , J * φ = ϕ, J * (δ N φ) = 0 on M (7)
admits a solution φ (which is also a p-form) on N (see Lemma 3.4.7 in [START_REF] Schwarz | Hodge decomposition-A method for solving boundary value problems[END_REF]). The solution is unique, up to a Dirichlet harmonic field. In particular, it is proved in [1, Lemma 3.1] that any solution φ is co-closed on N and that d N φ ∈ H p+1 (N ).

Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary

In this section, we will prove Theorem 1.1. For this, we consider a compact Riemannian manifold N whose boundary M carries a Riemannian flow. We start by establishing two lemmas that will be crucial for the estimate. In the first one and by using the boundary value problem (7), we will see that any basic closed p-form on M can be extended to a closed and co-closed p-form on N . In the second lemma, we will prove that Inequality (4) can be expressed in another way when one restricts to basic differential forms. First, we have:

Lemma 3.1 Let (N n+2 , g) be a Riemannian manifold with W [p+1] N
≥ 0 for some 1 ≤ p ≤ n. Assume that the boundary M carries a Riemannian flow given by a unit vector field ξ. Let ϕ be any basic closed p-form. If σ n+1-p (M ) > 0, then φ is closed and co-closed on N .

Proof. As we mentioned in Section 2 the form φ, solution of the problem [START_REF] Gallot | Sur la première valeur propre du p-spectre pour les variétés à opérateur de courbure positif[END_REF], is co-closed on N and the (p + 1)-form ω := d N φ is also closed and co-closed on N . We then apply the Reilly formula for the form ω to get

0 = N |∇ N ω| 2 v g + N W [p+1] N (ω), ω v g + M S [n+1-p] (J * * N ω), J * * N ω v g .
Here we used the fact that J * ω = d M ϕ = d b ϕ = 0, since the form ϕ is a basic closed form. Now using Inequality (4), we have at any point of M that

S [n+1-p] (J * * N ω), J * * N ω ≥ σ n+1-p (M )|J * * N ω| 2 = σ n+1-p (M )|ν ω| 2 = σ n+1-p (M )|ω| 2 .
Therefore from the fact that W

[p+1] N ≥ 0 and σ n+1-p (M ) > 0, we deduce that ω vanishes on M and is also parallel on N . Hence it vanishes everywhere and thus φ is closed.

Next, we state the second lemma: Lemma 3.2 Let (N n+2 , g) be a Riemannian manifold with boundary M . Assume that M carries a Riemannian flow given by a unit vector field ξ. For any basic p-form ϕ on M , we have

S [p] ϕ, ϕ ≥ (σ p+1 (M ) -g(S(ξ), ξ))|ϕ| 2 .
Proof. By using Equality (5) for X = ξ, we can write

S [p+1] (ξ ∧ ϕ), ξ ∧ ϕ = S(ξ) ∧ ϕ, ξ ∧ ϕ + ξ ∧ S [p] ϕ, ξ ∧ ϕ = ξ (S(ξ) ∧ ϕ), ϕ + ξ (ξ ∧ S [p] ϕ), ϕ = g(S(ξ), ξ)|ϕ| 2 + S [p] ϕ, ϕ -ξ ∧ (ξ S [p] ϕ), ϕ = g(S(ξ), ξ)|ϕ| 2 + S [p] ϕ, ϕ ,
where the term ξ ∧ (ξ S [p] ϕ), ϕ vanishes because ϕ is basic. We finally finish the proof of the lemma by using Inequality (4) and the fact that |ξ ∧ ϕ| = |ϕ|.

Since ∆ b commutes with d b and δ b , the space of closed (resp. co-closed) basic forms is also preserved. For this, we denote λ 1,p (resp. λ 1,p ) as the first eigenvalue of the basic Laplacian operator restricted to closed (resp. co-closed) p-forms. We then have that the first eigenvalue λ 1,p of ∆ b is equal to λ 1,p = min(λ 1,p , λ 1,p ) and that λ 1,p = λ 1,n-p [START_REF] Gallot | Sur la première valeur propre du p-spectre pour les variétés à opérateur de courbure positif[END_REF]. These two facts come from the basic Hodge-de Rham decomposition and basic Poincaré duality (the flow is assumed to be minimal) [START_REF] Kamber | De-Rham Hodge theory for Riemannian foliations[END_REF][START_REF] Kamber | Duality theorems for foliations[END_REF][START_REF] Richardson | The basic Laplacian of a Riemannian foliation[END_REF]. In [START_REF] Jung | Transverse conformal Killing forms and a Gallot-Meyer theorem for foliations[END_REF], the authors established an estimate for the first eigenvalues λ 1,p and λ 1,p à la Gallot-Meyer for Riemannian foliations; for this purpose they assume that the normal curvature (i.e. the one of the normal bundle) is bounded from below by some constant. They also prove that if the estimate is attained for λ 1,1 (i.e. on closed 1-forms), the foliation is transversally isometric to the action of a discrete subgroup of O(n) acting on the sphere (see [START_REF] Lee | Lichnerowicz and Obata theorems for foliations[END_REF] for the definition) where n is the rank of the normal bundle. Now, we are ready to prove Theorem 1.1:

Proof of Theorem 1.1. Let ϕ be any basic 1-eigenform for the basic Laplacian that is closed. From Lemma 3.1, it admits an extension φ which is closed and co-closed on N . The Reilly formula applied then to the 1-form φ gives, under the curvature assumption and the use of Lemma 3.2 for the eigenform ϕ, that

0 ≥ 2 M ν φ, δ M ϕ v g + σ 2 (M ) M |ϕ| 2 v g - M g(S(ξ), ξ)|ϕ| 2 v g + σ n+1 (M ) M |ν φ| 2 v g .
Now from the pointwise inequality |ν φ + 1 σ n+1 (M ) δ M ϕ| 2 ≥ 0, the above one can be reduced to the following

M |δ M ϕ| 2 v g ≥ σ 2 (M )σ n+1 (M ) M |ϕ| 2 v g -σ n+1 (M ) sup M g(S(ξ), ξ) M |ϕ| 2 v g . (8)
We notice here that σ n+1 (M ) is positive since we have pointwise σ n+1 ≥ σ n > 0. Using the relation

δ b = δ M on basic 1-forms [19, Prop.2.4], Inequality (8) implies λ 1,1 M |ϕ| 2 v g = M |δ b ϕ| 2 v g ≥ σ n+1 (M ) σ 2 (M ) -sup M g(S(ξ), ξ) M |ϕ| 2 v g ,
which finishes the proof of the theorem.

The equality case

In this section, we study the limiting case of Inequality (2). We will prove that, under the condition that all the principal curvatures are non-negative, the boundary has to be η-umbilical (see Lemma 4.1). This means that the second fundamental form vanishes along ξ and is equal to a multiple of the identity in the direction of Q. We will also show that the O'Neill tensor defining the flow vanishes; this is equivalent to say that the normal bundle is integrable (see Lemma 4.2). Moreover, we will see that the extension ξ of the vector field ξ (coming from the problem ( 7)) is parallel on the whole manifold. This will allow us to classify all manifolds for which the estimate (2) is optimal.

First, we begin to prove that the boundary is η-umbilical.

Lemma 4.1 Under the assumptions of Theorem 1.1 with σ 1 (M ) ≥ 0, if the equality is attained in (2), then M is η-umbilical in N , i.e. S(ξ) = 0 and S(X) = ηX for X orthogonal to ξ.

Proof. The equality is attained in (2) if and only if λ 1,1 = σ n+1 (M ) σ 2 (M ) -g(S(ξ), ξ) . In this case φ is parallel on N and the functions σ 2 , σ n+1 and g(S(ξ), ξ) are constant. Moreover, we have (see [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF], formulas ( 15) and ( 23)),

           ∇ M X ϕ = g( φ, ν)S(X), for all X ∈ Γ(T M ), δ M ϕ = -(n + 1)Hν φ d M (ν φ) = -S(ϕ). (9)
Hence we can write g( φ, ν)g(S(ξ), ξ)

= g(∇ M ξ ϕ, ξ) = -g(ϕ, κ) = 0. Recall here that the flow is assumed to be minimal. As g(S(ξ), ξ) is constant, it is either zero or g( φ, ν) = 0. But if g( φ, ν) = 0, we obtain from the first equation in ( 9) that ∇ M X ϕ = 0 for all X ∈ Γ(T M ) and therefore ∆ b ϕ = λ 1,1 ϕ = 0, which gives a contradiction. Thus we deduce that g(S(ξ), ξ) = 0. Now, since 0 ≤ η 1 = σ 1 ≤ g(S(ξ), ξ) = 0 one gets from the one hand that η 1 = 0. On the other hand and in order to prove that S(ξ) = 0, we consider an orthonormal basis {f 1 , . . . , f n+1 } of eigenvectors of S associated with the eigenvalues η 1 , . . . , η n+1 . By writing the vector field ξ in this orthonormal frame as ξ = n+1 i=1 a i f i for some real functions a i , one gets that

0 = g(S(ξ), ξ) = n+1 i=2 a 2 i η i .
Recall that all the η i 's are non-negative for i = 2, • • • , n+1. If η 2 = 0, then λ 1,1 = σ 2 σ n+1 = 0 which is impossible. Finally, we deduce that all the a i 's are zero for i ≥ 2 which means that S(ξ) = 0.

To prove the other part of the lemma, we use (9) to write

S(ϕ) = 1 (n + 1)H d M (δ M ϕ) = 1 σ n+1 λ 1,1 ϕ = σ 2 -g(S(ξ), ξ) ϕ = σ 2 ϕ.
By differentiating this equation along any vector field X ∈ Γ(T M ) and using again the first equation in [START_REF] Hijazi | Extrinsic Killing spinors[END_REF], we obtain from the one hand

∇ M X S(ϕ) = σ 2 g( φ, ν)S(X). (10)
On the other hand, we write

∇ M X S(ϕ) = (∇ M X S)(ϕ) + S(∇ M X ϕ) = (∇ M ϕ S)(X) + R N (ϕ, X)ν + g( φ, ν)S 2 (X) = ∇ M ϕ (S(X)) -S(∇ M ϕ X) + R N (ϕ, X)ν + g( φ, ν)S 2 (X), (11)
where in the second equality we used the Gauss-Codazzi equation. Now by combining ( 10) and ( 11), we find the following identity

∇ M ϕ S(X) -S(∇ M ϕ X) + R N (ϕ, X)ν + g( φ, ν)S 2 (X) = σ 2 g( φ, ν)S(X).
Taking the scalar product of the last equation with X and tracing over an orthonormal frame of T M , we obtain after using the fact that σ n+1 = (n + 1)H is constant

-g( φ, ν)Ric N (ϕ, ν) = (σ 2 σ n+1 -|S| 2 )g( φ, ν) 2 . (12)
It is easy to see that the inequality σ 2 σ n+1 -|S| 2 ≤ 0 holds with equality if and only if

η 2 = • • • = η n+1 .
To check the sign of the l.h.s. of ( 12), we use the fact that Ric N ( φ, φ) = 0 (the vector field φ is parallel) with the decomposition φ = g( φ, ν)ν + ϕ along the boundary. This allows to deduce that

g( φ, ν) 2 Ric N (ν, ν) + 2g( φ, ν)Ric N (ϕ, ν) + Ric N (ϕ, ϕ) = 0. ( 13
)
Hence, the second term in the l.h.s. of ( 13) is non-positive as a consequence of the non-negativity of the curvature. That mainly means both terms in (12) vanish, which ends the proof.

In the following, we will prove that when the equality of the estimate is realized, the O'Neill tensor vanishes. The proof relies on the fact that the eigenform ϕ (up to its norm) defines a geodesic vector field and therefore the O'Neill tensor satisfies a differential equation along those geodesics. It turns out that the solution of such differential equation, that we find explicitly, blows up at some limit. This contradicts the compactness of the boundary.

Lemma 4.2 Under the assumptions of Theorem 1.1 with σ 1 (M ) ≥ 0, if the equality is attained in (2), the flow is then a local product, that is the O'Neill tensor h is equal to 0.

Proof. First, note that using (9), we have

X(g( φ, ν)) = -g(S(ϕ), X) = -ηg(ϕ, X), for X ∈ Γ(T M ). ( 14
)
From the one side, the curvature on M of the eigenform ϕ when applied to ξ and Y ∈ Γ(Q) gives after using the η-umbilicity of the boundary and the first identity in ( 9) that

R M (ξ, Y )ϕ = ∇ M ξ ∇ M Y ϕ -∇ M Y ∇ M ξ ϕ -∇ M [ξ,Y ] ϕ = ∇ M ξ (g( φ, ν)ηY ) - n i=1 g([ξ, Y ], e i )g( φ, ν)ηe i = ηξ(g( φ, ν))Y + ηg( φ, ν)∇ M ξ Y - n i=1 g(∇ M ξ Y -∇ M Y ξ, e i )g( φ, ν)ηe i = -ηg(ϕ, S(ξ))Y + ηg( φ, ν)∇ M ξ Y - n i=1 g(∇ M ξ Y, e i )g( φ, ν)ηe i + n i=1 g(h(Y ), e i )g( φ, ν)ηe i = ηg( φ, ν)g(∇ M ξ Y, ξ)ξ + ηg( φ, ν)h(Y ) = ηg( φ, ν)h(Y ), (15)
where {e 1 , . . . , e n } is a local orthonormal frame of Γ(Q). From the other side, by differentiating the relation h(ϕ) = 0, that we can get from ( 3) and ( 9), along any vector field Y orthogonal to ξ we obtain

∇ M ϕ ∇ M Y ξ + R M (Y, ϕ)ξ + ∇ M [Y,ϕ] ξ = 0. This implies that ∇ M ϕ h(Y ) + R M (Y, ϕ)ξ + n i=1 g(∇ M Y ϕ -∇ M ϕ Y, e i )∇ M e i ξ = 0.
Taking now the scalar product of the last identity with h(Y ) and tracing over an orthonormal frame of Γ(Q), we deduce after using again (9) that

1 2 ϕ(|h| 2 ) - n j=1
R M (ξ, h(e j ), ϕ, e j ) + ηg( φ, ν)|h| 2 = 0.

Combining the last equation with [START_REF] Li | A sharp comparison theorem for compact manifolds with mean convex boundary[END_REF], we finally get the relation

ϕ(|h| 2 ) = -4ηg( φ, ν)|h| 2 . ( 16
)
In the following, we will prove that the vector field Y = ϕ |ϕ| is geodesic and that the O'Neill tensor satisfies a differential equation along the geodesic curves of this vector field. For this, we set

U = {x ∈ M | ϕ(x) = 0}. Clearly, the set U is open and dense in M . To show that Y is geodesic, we compute ∇ M Y Y = ϕ( 1 |ϕ| )ϕ + 1 |ϕ| ∇ M ϕ ϕ 1 |ϕ| = -ϕ(|ϕ|) |ϕ| 2 ϕ + 1 |ϕ| g( φ, ν)ηϕ 1 |ϕ| . Since ϕ(|ϕ| 2 ) = 2g(∇ M ϕ ϕ, ϕ) = 2g( φ, ν)η|ϕ| 2 , then |ϕ|ϕ(|ϕ|) = g( φ, ν)η|ϕ| 2 . ( 17
)
That means ∇ M Y Y = 0. Let us denote by c(t) a maximal geodesic of Y starting at some point x ∈ U and consider the function f = g( φ, ν) • c. Then a direct computation of the first derivative gives that ḟ

(t) = g c(t) (df, Y ) = -ηg c(t) (ϕ, Y ) = -η|ϕ| c(t) < 0. The second derivative is equal to f = g(∇ M Y df, Y ) = -ηg( φ, ν)g(SY, Y ) = -η 2 f.
Solving this last differential equation, we get f (t) = A cos(ηt) + B sin(ηt), where A and B are arbitrary constant. In the following, we will find the differential equation satisfied by l := |h| 2 • c and will try to solve it explicitly. In fact, the first derivative of l is equal to

l = Y (|h| 2 ) = ϕ |ϕ| (|h| 2 ) (16) = -4η g( φ, ν) |ϕ| l, (18)
and its second derivative is

l = g(∇ M Y dl, Y ) = 1 |ϕ| ϕ( 1 |ϕ| ϕ(|h| 2 )). ( 19
)
But, we have

ϕ 1 |ϕ| ϕ(|h| 2 ) = ϕ 1 |ϕ| ϕ(|h| 2 ) + 1 |ϕ| ϕ ϕ(|h| 2 ) (16) = -ϕ(|ϕ|) |ϕ| 2 -4ηg( φ, ν)|h| 2 + 1 |ϕ| (-4η)ϕ(g( φ, ν)|h| 2 ) (17) = 4η 2 g( φ, ν) 2 |ϕ| |h| 2 - 4η |ϕ| ϕ(g( φ, ν))|h| 2 + g( φ, ν)ϕ(|h| 2 ) (14),( 16
) = 4η 2 g( φ, ν) 2 |ϕ| |h| 2 - 4η |ϕ| -η|ϕ| 2 |h| 2 -4ηg( φ, ν) 2 |h| 2 = 20η 2 g( φ, ν) 2 |ϕ| |h| 2 + 4η 2 |ϕ||h| 2 .
Hence we deduce that l = 20η 2 g( φ, ν)

2 |ϕ| 2 + 4η 2 l. (20)
Combining now equations ( 18) and ( 20), we deduce l l = 5 l2 4l 2 + 4η 2 . Solving this last equation, we obtain ll = 4η tan(ηt + E) and thus l = D cos 4 (ηt + E)

where E is an arbitrary constant and D is a non-negative constant. In conclusion, we find that:

       f (t) = A cos(ηt) + B sin(ηt) with ḟ < 0 l(t) = D cos 4 (ηt + E) , D ≥ 0.
Note also that l 4ηl = tan(ηt + E)

(18) = ηf (t) ḟ (t) = - A cos(ηt) + B sin(ηt) A sin(ηt) -B cos(ηt) . ( 21
)
Those equations are defined over the interval I = c -1 (U ). Let c : R → M be the geodesic extension of the curve c to all R (which exists because M is compact) and consider the function f = g( φ, ν)•c.

Clearly, we have that c -1 (U ) = c-1 (U ) and that f = f on I. At all points outside I, the function f is equal to a constant, because of the formula | φ| 2 = |ϕ| 2 + f 2 . Recall here that the form φ is parallel and thus is of constant norm. Therefore, we get that

f (t) =    Acos(ηt) + Bsin(ηt) t ∈ I const. otherwise.
But it is easy to check that the function f is not of class C 2 . Hence ϕ(c(t)) = 0 and we deduce that c = c : R → U.

Let us consider J the maximal interval containing 0 on which ḟ < 0 (it will be the same as for the function l). We shall prove that on this interval, the function l tends to +∞ which contradicts the fact that it is a bounded function, as M is compact. We first notice that tan E = A B (replace t = 0 in Equation ( 21)) and that B < 0 (take t = 0 in ḟ ). Depending on the sign of A, we distinguish the following cases:

• Case where A = 0 : In this case, the interval J is equal to ] -π 2η , π 2η [ and E = kπ. Hence, as t → ± π 2η , the function l(t) = D cos 4 (ηt) tends to +∞ which is a contradiction.

• Case where A > 0 : The interval J is ]α, β[, where α and β are given by

α = 1 η arctan B A , -π 4η < α < 0 and β = 1 η (arctan B A + π), 3π 4η < β < π η .
At these values of α and β, the derivative of f vanishes while f does not vanish. From ( 21), we get tan(ηt + E) → ∞ as t goes to α or to β, i.e. l(t) tends to infinity which is also a contradiction.

• Case where A < 0 : We can use a similar argument as in the previous case. Now, we have all the materials to prove Theorem 1.2:

Proof of Theorem 1.2. In order to prove that the manifold N is a local product, we need first to show that the vector field ξ defining the flow can be extended to a unique parallel vector field ξ on N which is orthogonal to ν. For this purpose, we proceed as in [START_REF] Chami | Rigidity results for spin manifolds with foliated boundary[END_REF].

Let ξ be the solution of the boundary problem (7) associated with the vector field ξ on M . Let us consider the 2-form ω := d N ξ which is clearly closed and co-closed on N . Its restriction to the boundary is J * ω = d M ξ = 0, since ξ is parallel on M. Therefore, the Reilly formula applied to ω reduces to

0 = N |∇ N ω| 2 + W [2] N (ω), ω v g + (n + 1) M H|ν ω| 2 v g - M S(ν ω), ν ω v g . (22)
As the second fundamental form is equal to zero in the direction of ξ and to ηId in the orthogonal direction to ξ, we find for any 1-form β on M that S(β)(X) = ηβ(X) for X ⊥ ξ and S(β)(ξ) = 0.

Hence, S(ν ω), ν ω = η n i=1 (ν ω)(e i ) 2 , where {ξ, e 1 , • • • , e n } is a local orthonormal frame of T x M . Using the fact that (n + 1)H = nη, Equation [START_REF] Schwarz | Hodge decomposition-A method for solving boundary value problems[END_REF] gives

0 = N |∇ N ω| 2 + W [2] N (ω), ω v g + (n -1)η M |ν ω| 2 v g + η M (ν ω)(ξ) 2 v g .
Using the curvature assumption on N , we deduce that ω is parallel on N and ν ω = 0 on M . But at each point of M , we have |ω| 2 = |ν ω| 2 + |J * ω| 2 = 0. Therefore ω vanishes on N ; which means that ξ is closed. Applying now the Stokes' formula

N d N α, β v g = N α, δ N β v g - M J * α, ν β
to α = δ N ξ and β = ξ, we deduce that ξ is co-closed on N . Finally, we use again the Reilly formula to the vector field ξ to get

0 = N |∇ N ξ| 2 v g + Ric N ( ξ), ξ v g + (n + 1) M H|ν ξ| 2 v g .
This implies that ξ is parallel on N and ν ξ = 0 on M .

In the sequel, we follow the same proof as in [START_REF] Chami | Rigidity results for spin manifolds with foliated boundary[END_REF]. As ξ is parallel on N , we consider a connected integral submanifold N 1 of the bundle (R ξ) ⊥ , where the orthogonal is taken in N . It is also straightforward to see that N 1 is the quotient of a totally geodesic hypersurface N 1 of the universal cover N (which is complete) of N . In particular, the manifold N 1 is complete as being a level hypersurface of the function f defined on N by d N f = ξ (recall here that d N ξ = 0). From the fact that the universal cover is a local isometry, we deduce that N 1 is complete. On the other hand, the boundary of N 1 is a totally geodesic hypersurface in M with a normal vector field ξ and it is totally umbilical in N 1 since the second fundamental form is equal to -∇ N 1 X ν = -∇ N X ν = ηX. Hence by the Gauss formula, and for any X ∈ Γ(T ∂N 1 ), the Ricci curvature of N 1 is equal to

g(Ric ∂N 1 X, X) = n i=1 g(R M (X, e i )e i , X) = n i=1 g(R N (X, e i )e i , X) -g(S 2 (X), X) + (n + 1)Hg(S(X), X) = (n -1)η 2 |X| 2 + n i=1 g(R N (X, e i )e i , X) ≥ (n -1)η 2 |X| 2 ,
where the last inequality comes from the fact that the curvature operator on N is positive. Here {e i } i=1,••• ,n is a local orthonormal frame of Γ(Q). Therefore by Myers's theorem, we deduce that ∂N 1 is compact which with the rigidity result in [START_REF] Li | A sharp comparison theorem for compact manifolds with mean convex boundary[END_REF]Thm. 1.1] allows to say that N 1 is compact.

On the other hand, we have from Equations ( 9) that ϕ = -1 η d M (ν φ) which means that it is d Mexact and thus d ∂N 1 -exact, as ∂N 1 is totally geodesic in M and both ϕ and ν φ are basic. Moreover, the basic form ϕ is an eigenform of the Laplacian on ∂N 1 , that is ∆ ∂N 1 ϕ = ∆ b ϕ = λ 1,1 ϕ. Therefore, if we denote by λ ∂N 1 1,1 the first eigenvalue of ∆ ∂N 1 restricted to exact forms on N 1 and by σ p (∂N 1 ) the p-curvatures of ∂N 1 into the compact manifold N 1 , we get from the main estimate in [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF]Thm. 5] (see Inequality [START_REF] Belishev | Dirichlet to Neumann operator on differential forms[END_REF]) that

nη 2 = σ 1 (∂N 1 )σ n (∂N 1 ) ≤ λ ∂N 1 1,1 ≤ λ 1,1 = σ 2 (M )σ n+1 (M ) = nη 2 .
Hence the equality is attained in the estimate of Raulot and Savo [16, Thm. 1] and therefore N 1 is isometric to the Euclidean closed ball B . Finally, by the de Rham theorem, the manifold N is isometric to R × B and N is the quotient of the Riemannian product R × B by its fundamental group. As π 1 (N ) embeds into π 1 (M ) (any isometry of B fixing pointwise S n is the identity map), N is isometric to Γ \ R × B , for some fixed-point-free cocompact discrete subgroup Γ ⊂ R × SO n+1 .

Examples 4.1 First, we point out that when the transversal Ricci curvature is positive, the first basic cohomology group is zero [START_REF] Hebda | Curvature and focal points in Riemannian foliations[END_REF]. In this case, the first positive eigenvalue on basic closed forms is equal to the first positive eigenvalue on basic exact forms which is also equal to the first positive eigenvalue of the basic Laplacian on functions. This is the case for the manifold M = Γ \ R × S n with the foliation given by the vector field ∂t.

In general, the limiting case of Inequality (2) is not characterized by M being isometric to Γ \ R × S n and N isometric to Γ \ R × B , where B is the unit closed ball in R n+1 . To see this, we consider the action of Γ on R × S 3 by (0, p) → (1, -p). For this example, the lower bound of Inequality (2) is equal to 3 which is the first eigenvalue on S 3 associated to a homogeneous harmonic polynomial on R 4 of degree 1, as an eigenfunction. But a polynomial of degree 1 is not invariant by the action of Γ. That means the first eigenvalue is strictly bigger than 3 (it is in fact equal to 8) and hence the equality case in (2) is not attained. Recall here that the eigenvalues on the round sphere S n are given by l(l + n -1) for l ≥ 0. This example can be generalized to any quotient Γ \ R × S n where n is odd because for the odd-dimensional sphere S n we don't necessarily have any fixed directions by an element of SO(n + 1), so the first invariant eigenvalue could be higher than n.

However, for the even-dimensional sphere, any element of SO(n + 1) must always have a direction that is fixed (eigenvector for the eigenvalue 1), say v. The function f (x) = v • x is a linear function with eigenvalue n that is invariant by that element of SO(n + 1). Thus, for n even and for this special discrete group Γ, the equality case is characterized by M. Let us give another example. Consider the quotient of M = Γ \ R × S 2 where Γ acts on R × S 2 by (z, θ, 0) → (z, θ +α, 1). Here α denotes an irrational multiple of 2π. In this case, the basic Laplacian is given by ∆ b = -(1 -z 2 )∂ 2 z + 2z∂ z and its eigenvalues are given by l(l + 1) for l ≥ 0 [START_REF] Richardson | The asymptotics of heat kernels on Riemannian foliations[END_REF]. In this case the first positive eigenvalue of ∆ b is 2 and hence the equality case in (2) is satisfied.

Rigidity results on manifolds with foliated boundary

In this section, we study the case where the Riemannian flow carries some special forms (that we shall call them special Killing forms [START_REF] Semmelmann | Conformal Killing forms on Riemannian manifolds[END_REF][START_REF] Tachibana | On a Riemannian space admitting more than one Sasakian structures[END_REF]) in order to derive rigidity results on manifolds with foliated boundary arising from the equality case of our main estimate. On a Riemannian manifold (M, g), a special Killing p-form ω is a co-closed form satisfying for all X ∈ Γ(T M ) the relations

∇ X ω = 1 p + 1 X dω and ∇ X dω = -λ(p + 1)X ∧ ω,
where λ is a non-negative constant (see also [START_REF] Jung | Transverse conformal Killing forms and a Gallot-Meyer theorem for foliations[END_REF]). Killing forms have been studied in many papers.

For example, we quote the work of [START_REF] Semmelmann | Conformal Killing forms on Riemannian manifolds[END_REF]Thm. 4.8] where it is shown that there is a one-to-one correspondence between those forms and parallel forms constructed on the cone of the underlying manifold. This result has led to a complete classification of compact simply connected manifolds carrying such forms. For Riemannian flows (or in general Riemannian foliations), one can adapt the same definition of special Killing forms to the set-up of basic forms. Here, we recall that ∇ is the transversal Levi-Civita connection defined in Section 2. As for the general case, one can show that a basic special Killing p-form is a co-closed eigenform of the basic Laplacian corresponding to the eigenvalue λ(p + 1)(n -p) where n is the rank of Q.

In the following, we will consider a compact manifold N whose boundary carries a basic special Killing p-form. Depending on the sign of the term g(S(ξ), ξ), we will characterize the boundary as being the product S 1 × S n (since it admits basic special Killing forms) and will see that, under some assumptions on the principal curvature, the manifold N is isometric to the product of S 1 with the unit closed ball. This comes in fact from the equality case of the estimate (2).

We first have: Here " * b " is the basic Hodge star operator [START_REF] Ph | Foliations on Riemannian manifolds[END_REF] which commutes with the basic Laplacian as the flow is minimal. Hence, we get the estimate λ 1,1 ≤ n. In order to get the lower bound from Theorem 1.1, we need to check the condition on σ n . For this, we consider the functions θ i = σ i+1 -η 1 for all i = 1, • • • , n. We then have

σ n = θ n-1 + η 1 ≥ (n -1)θ 1 + η 1 ≥ (n -1)(1 -η 1 ) + η 1 = (n -1) -(n -2)η 1 > 0,
since η 1 ≤ g(S(ξ), ξ) ≤ 0. Moreover, we compute

σ n+1 = θ n + η 1 ≥ nθ 1 + η 1 ≥ n -η 1 (n -1) ≥ n.
Therefore, we deduce that λ 1,1 ≥ n from the main estimate in Theorem 1.1 and the equality case is realized (σ 1 = 0 because η 1 = 0). This finishes the proof as a consequence of the equality case.

Using this last result, we provide a proof of Corollary 1.3:

Proof of Corollary 1.3. As a consequence of Corollary 5.1, the manifold N must be isometric to the quotient of R × B by some discrete free fixed point cocompact subgroup Γ. Since M is assumed to be isometric to S 1 × S n , then Γ's action is trivial on the S n factor, that means that N is isometric to S 1 × B .

In the following, we will relax the condition on σ 2 by assuming that the sectional curvature of N vanishes along planes on T M .

Corollary 5.2 Let N be a (n + 2)-dimensional compact manifold with non-negative curvature operator. Assume that M = S 1 × S n with n ≥ 3, the sectional curvature K N of N vanishes on M, the mean curvature H > 0 and g(S(ξ), ξ) ≤ 0. Then, the manifold N is isometric to S 1 × B . From the assumption on g(S(ξ), ξ), we deduce that from the one hand that S(ξ) = 0. On the other hand, we consider an orthonormal basis {ξ, f 2 , . . . , f n+1 } of eigenvectors of S associated to the eigenvalues 0, η 2 , . . . , η n+1 . For i = j, we get K M (f i , f j ) = K N (f i , f j ) + η i η j , which gives η i η j = 1 for all i = j. We thus conclude that all η i 's are constants, equal to 1 or -1.

Since H > 0 then η i = 1 and hence σ 2 = 1. The result is deduced by using Corollary 1.3.

This last result could be compared to Theorem 2 in [START_REF] Xia | Rigidity of compact manifolds with boundary and non-negative Ricci curvature[END_REF] where the author characterized the round sphere as the boundary of a compact manifold under similar assumptions. We finally end this section by stating similar results as before: Here we used the fact that η 1 ≤ g(S(ξ), ξ). Moreover,

σ n+1 = θ n + η 1 ≥ nθ 1 + η 1
≥ n(1 + g(S(ξ), ξ) -η 1 ) + η 1

= n + ng(S(ξ), ξ) -(n -1)η 1

≥ n + g(S(ξ), ξ) ≥ n.

This finishes the proof of the corollary.

Finally, a similar proof can be done to deduce Corollary 1.4 as the one of Corollary 1.3.

We end this paper by the following comment: As we mentioned in the introduction, the case where p > 1 will be treated in another work, since the estimate would not be of the same type. Even though Lemmas 3.1 and 3.2 in Section 3 are true for any p, the relations between the codifferentials δ M and δ b involves terms that mainly depend on the 2-form Ω for p > 1 [START_REF] Richardson | The basic Laplacian of a Riemannian foliation[END_REF]. In this situation, the equality case would not be of the same technique.

  M and hence one may set d b := d M | Ω B (M ) . When M is compact, we denote by δ b the L 2 -adjoint of d b and define the basic Laplacian as being ∆ b = d b δ b + δ b d b .

Corollary 5 . 1

 51 Let N be a (n + 2)-dimensional compact manifold with non-negative curvature operator. Assume that the boundary M carries a minimal Riemannian flow such that g(S(ξ), ξ) ≤ 0 and also admits a basic special Killing (n -1)-form. If the inequality σ 2 (M ) ≥ 1 holds, the manifoldN is isometric to Γ \ R × B .Proof. Let ϕ be a basic special Killing (n -1)-form on M . Then * b ϕ is a basic closed 1-eigenform for the basic Laplacian, that is ∆ b ( * b ϕ) = n( * b ϕ).

  Proof. For X, Y ∈ Γ(T M ) with |X| = |Y | = 1 and g(X, Y ) = 0, the sectional curvatures of M and N are related byK M (X, Y ) = K N (X, Y ) + g(S(X), X)g(S(Y ), Y ) -g(S(X), Y ) 2 .By applying this formula to X = ξ ∈ T S 1 and Y = e i ∈ T S n , where {ξ, e 1 , . . . , e n } is an orthonormal frame of T M , we obtain (n + 1)Hg(S(ξ), ξ) -|S(ξ)| 2 = 0.

Corollary 5 . 3 1 ≥ 1 ≥

 5311 Let N be a (n + 2)-dimensional compact manifold with non-negative curvature operator. Assume that the boundary M carries a minimal Riemannian flow such that g(S(ξ), ξ) ≥ 0 and also admits a basic special Killing (n -1)-form.If the inequality σ 2 (M ) ≥ 1 + sup M g(S(ξ), ξ) holds, the manifold N is isometric to Γ \ R × B .Proof. We follow the same proof as in Corollary 5.1. It is sufficient to prove that σ n > 0 and that σ n+1 ≥ n. As before, we consider the functions θ i = σ i+1 -η 1 to computeσ n = θ n-1 + η 1 ≥ (n -1)θ 1 + η (n -1)(1 + g(S(ξ), ξ) -η 1 ) + η 1= (n -1) + (n -1)g(S(ξ), ξ) -(n -2)η (n -1) + g(S(ξ), ξ) > 0.

Acknowledgment: We would like to thank Nicolas Ginoux and Ken Richardson for many enlighting discussions and for pointing us a mistake in a previous version of the paper. The first two named authors were supported by a fund from the Lebanese University. The second named author would like to thank the Alexander von Humboldt Foundation for its support.


Fida El Chami
email: fchami@ul.edu.lb

Georges Habib
email: ghabib@ul.edu.lb

Ola Makhoul
email: ola.makhoul@ul.edu.lb

Roger Nakad
email: rnakad@ndu.edu.lb

Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary, part II

Keywords: Riemannian flow, manifolds with boundary, basic Laplacian, eigenvalue, second fundamental form, O'Neill tensor, basic Killing forms, rigidity results Mathematics Subject Classification: 53C12, 53C24, 58J50, 58J32

On a compact Riemannian manifold N whose boundary is endowed with a Riemannian flow, we gave in [4] a sharp lower bound for the first non-zero eigenvalue of the basic Laplacian acting on basic 1-forms. In this paper, we extend this result to the set of basic p-forms when p > 1. We then characterize the limiting case by showing that the manifold N is isometric to Γ \ R × B for some group Γ where B denotes the unit closed ball. As a consequence, we describe the Riemannian product S 1 × S n as the boundary of a manifold.

Introduction

On a compact Riemannian manifold (N n+2 , g) whose boundary M carries a minimal Riemannian flow given by a unit vector field ξ (see Section 2 for the definition of a Riemannian flow), we derived in [START_REF] Chami | Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary[END_REF] a sharp lower bound for the first non-zero eigenvalue λ 1,p of the basic Laplacian acting on (closed) basic p-forms on M when p = 1. This lower bound involves the q-curvatures σ q (1 ≤ q ≤ n + 1) of M [4, Thm. 1.1] which are defined as the sum of the first q principal curvatures assumed to be arranged in an increasing way. Here the word minimal means that the mean curvature of the integral curves of ξ is zero. The main tool to prove the estimate is the use of a suitable extension of a basic closed p-form to the whole manifold N (see Lemma 3.1) and the so-called Reilly formula established in [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF]. As a consequence of the limiting case, we got several rigidity theorems that characterize the flow as a local product and the boundary as an η-umbilical submanifold (see [START_REF] Chami | Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary[END_REF]Sec. 5]). These results can be seen as a foliated version of the work of S. Raulot and A. Savo in [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF].

In this paper, we generalize the results stated in [START_REF] Chami | Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary[END_REF] to any basic p-form when p > 1. Some of the techniques used in this paper are similar to those of p = 1 (Lemmas 3.1 and 3.2) but some new results and techniques are needed to extend the estimate and study the equality case (Lemmas 3.3, 4.2, 4.4 and 4.5). First, we prove Theorem 1.1 Let (N n+2 , g) be a compact Riemannian manifold with positive curvature operator. Assume that the boundary M is endowed with a minimal Riemannian flow given by a unit vector field ξ and that σ n+1-p (M ) > 0 for some 2 ≤ p ≤ n 2 . Then,

λ 1,p + 4[ n 2 ] sup M |Ω| 2 ≥ σ p+1 (M ) -sup M g(S(ξ), ξ) σ n+2-p (M ), ( 1
)
where Ω is the 2-form associated to the O'Neill tensor of the flow and S is the second fundamental form of M.

Inequality (1) differs from the one in [4, Thm. from the computation of the norm of the interior product of the O'Neill tensor with any basic p-form (see Lemma 3.3). As an example of manifolds that satisfy the assumptions of Theorem 1.1, one can consider the unit closed ball B in R n+2 where the boundary S n+1 is endowed with the Riemannian flow given by the Hopf fibration.

When equality is realized in [START_REF] Belishev | Dirichlet to Neumann operator on differential forms[END_REF], it turns out that Ω vanishes which yields to the following characterization:

Theorem 1.2 Under the same assumptions as in Theorem 1.1 with n ≥ 1 even, if σ 1 (M ) ≥ 0 and the equality case is realized, the manifold N is then isometric to the quotient Γ \ R × B for some fixed-point-free cocompact discrete subgroup Γ ⊂ R × SO n+1 , where B is the unit closed ball in R n+1 .

As a consequence, we describe manifolds whose boundaries are isometric to the Riemannian product S 1 × S n and get the following rigidity result:

Corollary 1.3 Let N be an (n + 2)-dimensional compact manifold with non-negative curvature operator. Assume that the boundary M is S 1 × S n and (n + 1 -p) sup M g(S(ξ), ξ) + 4c 2 [ n 2 ] does not change sign. If the inequality σ p+1 (M ) ≥ p holds, the manifold N is isometric to S 1 × B .

Riemannian flows and manifolds with boundary

Throughout this section, we recall the main ingredients of Riemannian flows defined on a manifold and the basic facts on manifolds with boundary (see [START_REF] Carrière | Flots riemanniens, structure transverse des feuilletages[END_REF][START_REF] Ph | Foliations on Riemannian manifolds[END_REF][START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF] and [START_REF] Chami | Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary[END_REF] for manifolds with foliated boundary). In the sequel, we will use the musical isomorphism between the tangent space of a Riemannian manifold and its dual. In particular, for any vector field X and a differential form ϕ, we write X ∧ ϕ for the form X * ∧ ϕ.

Let (M n+1 , g) be a Riemannian manifold and ξ be a smooth unit vector field on M defining the structure of a Riemannian flow on M . That is, the vector field ξ foliates the manifold M by its integral curves (called the leaves) in a way that those curves are locally equidistant [START_REF] Ph | Foliations on Riemannian manifolds[END_REF]. In other words, when one restricts the metric g to the bundle Q = ξ ⊥ it is then constant along the leaves. That means the relation L ξ g| ξ ⊥ = 0 holds. Equivalently to this last relation, the endomorphism h := ∇ M ξ (known as the O'Neill tensor [START_REF] O'neill | The fundamental equations of a submersion[END_REF]) defines a skew-symmetric tensor field on the bundle Q. Moreover, the normal bundle carries a covariant derivative ∇ (called transversal Levi-Civita connection) compatible with the induced metric g on Q [START_REF] Ph | Foliations on Riemannian manifolds[END_REF] that can be related with the Levi-Civita connection on M through the Gauss-type formulas: For all sections Z, W in Γ(Q), we have

   ∇ M Z W = ∇ Z W -Ω(Z, W )ξ, ∇ M ξ Z = ∇ ξ Z + h(Z) -κ(Z)ξ,
where κ := ∇ M ξ ξ is the mean curvature of the leaves and Ω(Z, W ) := g(h(Z), W ) is the differential 2-form associated to h. In all the paper, we will consider minimal Riemannian flows, that is κ = 0. Recall now that a basic form is a differential form ϕ on M such that ξ ϕ = 0 and ξ d M ϕ = 0. We will denote by Ω b (M ) the set of all such forms. Clearly, basic forms are preserved by the exterior derivative d M and therefore we can set From the spectral theory of transversally elliptic operators, the basic Laplacian has a discrete spectrum [START_REF] Kacimi | Stabilité du caractère kählérien transverse[END_REF][START_REF] Kacimi | Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications[END_REF].

d b := d M | Ω b (M ) .
Assume now that (N n+2 , g) is a Riemannian manifold of dimension n + 2 with boundary M . Recall that the shape operator (or the Weingarten tensor) is defined for all X ∈ Γ(T M ) as S(X) = -∇ N X ν where ∇ N is the Levi-Civita connection of N and ν is the inward unit nomal vector field along M . The Gauss-Codazzi equation is given for any X, Y ∈ Γ(T M ), by

(∇ M X S)(Y ) -(∇ M Y S)(X) = R N (Y, X)ν, (2)
where R N denotes the curvature tensor operator on N. At any point x ∈ M , we let η 1 (x), • • • , η n+1 (x) be the principal curvatures of M and arrange them so that holds for p ≤ q, and the equality is achieved if and only if either η 1 (x) = η 2 (x) = • • • = η q (x) or p = q. As mentioned in [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF], the Weingarten tensor admits a canonical extension to any p-form ϕ on M by the following:

η 1 (x) ≤ η 2 (x) ≤ • • • ≤ η n+1 (x
S [p] (ϕ)(X 1 , • • • , X p ) = p i=1 ϕ(X 1 , • • • , S(X i ), • • • , X p ),
for all vector fields X 1 , • • • , X p on M. The eigenvalues of S [p] are exactly the p-curvatures (that is,

p k=1 η i k , with i 1 < • • • < i p )
and that means the following inequality

S [p] (ϕ), ϕ ≥ σ p (M )|ϕ| 2 (3)
holds, where σ p (M ) is the infimum over M of the lowest p-curvatures σ p (x). We point out that this extension can be done for any symmetric tensor field on T M by the same definition. In particular, we will use it later for the tensor ∇ M X S for any X ∈ Γ(T M ). Now, we recall the Reilly formula established in [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF]Thm. 3]. Let J : M → N be the inclusion map of M into N and let J * be the pull-back of a form on N into M , that is J * is the restriction of differential forms on N to the boundary M. For any p-form α on N, the formula is the following

N (|d N α| 2 + |δ N α| 2 )v g = N |∇ N α| 2 v g + N W N p (α), α v g + 2 M ν α, δ M (J * α) v g + M B(α, α)v g
where v g is the volume element of g on N (also on M ) and W N p is the curvature term that appears in the Bochner-Weitzenböck formula for the Laplacian on N. The last term is defined by

B(α, α) = S [p] (J * α), J * α + S [n+2-p] (J * ( * N α)), J * ( * N α) = S [p] (J * α), J * α + (n + 1)H|ν α| 2 -S [p-1] (ν α), ν α .
We mention here that J * ( * N α) is equal (up to a sign) to * M (ν α) and the relation |J * α| 2 +|ν α| 2 = |α| 2 is true at any point of the boundary. We point out that the term W N p ≥ 0 when the curvature operator of N is non-negative.

Finally, the following boundary value problem will be of interest in our study. In fact, given any p-form ϕ on M , the solution of the system

   ∆ N φ = 0 on N , J * φ = ϕ, J * (δ N φ) = 0 on M (4)
is unique on N by Lemma 3.5.6 in [START_REF] Schwarz | Hodge decomposition-A method for solving boundary value problems[END_REF]. Moreover, the p-form φ is co-closed on N and d N φ belongs to the de Rham cohomology group H p+1 (N ) (see [START_REF] Belishev | Dirichlet to Neumann operator on differential forms[END_REF]Lemma 3.1] for more details).

Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary

In this section, we are going to prove Theorem 1.1. For this purpose, we need to state the following lemmas already proved in [START_REF] Chami | Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary[END_REF].

Lemma 3.1 [START_REF] Chami | Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary[END_REF] Let (N n+2 , g) be a compact Riemannian manifold with boundary M with W N p+1 ≥ 0 for some 1 ≤ p ≤ n. Assume that M carries a Riemannian flow defined by a unit vector field ξ such that σ n+1-p (M ) > 0. Given a non-zero basic closed p-form ϕ, the corresponding solution φ of the boundary value problem (4) is then closed and co-closed on N . Lemma 3.2 [START_REF] Chami | Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary[END_REF] Let (N n+2 , g) be a Riemannian manifold with boundary M . Assume that M carries a Riemannian flow given by a unit vector field ξ. For any basic p-form ϕ on M where 1 ≤ p ≤ n, we have

S [p] (ϕ), ϕ ≥ (σ p+1 (M ) -g(S(ξ), ξ))|ϕ| 2 .
This last estimate is optimal when N is isometric to S 1 × B , where B is the unit closed ball in R n+1 while Inequality (3) is not sharp in this example. Indeed, consider the flow by S 1 on the boundary M = S 1 × S n (i.e. the trivial fibration over S n ) and let ϕ be any non-zero p-form on S n . It is not difficult to check that S [p] (ϕ) = pϕ and that σ q = q -1 for any 1 ≤ q ≤ n + 1.

Next, we need to get an upper bound for the norm of the interior product of the 2-form Ω with any basic p-form. Indeed, Lemma 3.3 Let (N n+2 , g) be a Riemannian manifold with boundary M . Assume that M carries a Riemannian flow given by a unit vector field ξ. For any basic p-form ϕ with p ≥ 2, we have

|Ω ϕ| ≤ [ n 2 ] 1 2 |Ω||ϕ|. ( 5
)
For n even, the equality is realized if and only if Ω = 0.

Proof. As Ω is a skew-symmetric 2-tensor field on Q, we can always find a local orthonormal frame

{e i } of Γ(Q) such that Ω = [ n 2 ]
j=1 λ j e 2j-1 ∧ e 2j . Therefore, we compute

|Ω ϕ| = | [ n 2 ] j=1 λ j (e 2j-1 ∧ e 2j ) ϕ| ≤ [ n 2 ] j=1 |λ j ||e 2j-1 (e 2j ϕ)| ≤ [ n 2 ] j=1 |λ j ||ϕ| ≤ [ n 2 ] 1 2 |Ω||ϕ|.
Here we used the fact that |v ϕ| ≤ |v||ϕ| and the Cauchy-Schwarz inequality (in the last estimate). Assume now that the equality is realized, then either all the λ j s are of the same absolute value and there exists a j such that λ j = 0 (in this case, all the λ j 's are 0) or for all j, e 2j ∧ ϕ = 0 and e 2j-1 ∧ ϕ = 0. But for n even, the last statement just means that X ∧ ϕ = 0 for all X ∈ Γ(Q) and thus ϕ = 0. This leads to a contradiction; hence λ j = 0 for all j which yields to Ω = 0. Now, we have all the ingredients to prove Theorem 1.1:

Proof of Theorem 1.1. For any basic closed p-eigenform ϕ on M corresponding to the eigenvalue λ 1,p of the basic Laplacian, we associate its extension φ to N from Lemma 3.1. Applying now the Reilly formula to φ gives, under the curvature assumption and with the use of Lemma 3.2, the following

0 ≥ 2 M ν φ, δ M ϕ v g + σ p+1 (M ) M |ϕ| 2 v g - M g(S(ξ), ξ)|ϕ| 2 v g + σ n+2-p (M ) M |ν φ| 2 v g .
With the help of the pointwise inequality |ν φ+ 1 σ n+2-p (M ) δ M ϕ| 2 ≥ 0, the above one can be reduced to

M |δ M ϕ| 2 v g ≥ σ p+1 (M )σ n+2-p (M ) M |ϕ| 2 v g -σ n+2-p (M ) sup M (g(S(ξ), ξ)) M |ϕ| 2 v g . (6)
Now from the relation δ b = δ M -2Ω (ξ∧) on basic forms [12, Prop.2.4] and the estimate in Lemma 3.3, we compute

|δ M ϕ| 2 = |δ b ϕ| 2 + 4|Ω (ξ ∧ ϕ)| 2 + 4 δ b ϕ, Ω (ξ ∧ ϕ) = |δ b ϕ| 2 + 4|ξ ∧ (Ω ϕ)| 2 + 4 δ b ϕ, ξ ∧ (Ω ϕ) =0 since δ b ϕ is basic = |δ b ϕ| 2 + 4|Ω ϕ| 2 (5) ≤ |δ b ϕ| 2 + 4[ n 2 ]|Ω| 2 |ϕ| 2 .
Therefore after integrating over the manifold M, we get the desired result.

Remark 3.4 The assumptions in Theorem 1.1 on the curvature can be weakened. The positivity of the curvature operator can be replaced by the positivity of W N p and W N p+1 .

The equality case

This section is devoted to prove Theorem 1.2. In other words, we are going to study the limiting case of Inequality (1). We will show that, under some conditions, the second fundamental form vanishes along ξ and is equal to η Id in the direction of Q for some constant η, i.e. the boundary is η-umbilical. We will also prove that the O'Neill tensor defining the flow vanishes which is equivalent to the integrability of the normal bundle. Consequently, this allows to classify all manifolds on which Inequality (1) is optimal.

It is clear to see that when the equality is realized, the estimate in Lemma 3.3 is optimal which means that h = 0. On the other hand, the eigenform φ is parallel on N and σ p+1 , σ n+2-p and g(S(ξ), ξ) are constant on M . Moreover, we have the identity

δ M ϕ = -σ n+2-p ν φ. (7)
In particular, using the relations in [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF]Eq. (23) and Lemma 18 (ii)], we get for all X ∈ Γ(T M ),

                   ∇ M X ϕ = S(X) ∧ (ν φ) ∇ M X (ν φ) = -S(X) ϕ δ M ϕ = S [p-1] (ν φ) -σ n+1 ν φ d M (ν φ) = -S [p] (ϕ). (8)
By replacing Equation ( 7) (recall that σ n+2-p is constant) in the last two equations in [START_REF] Jung | Transversal conformal Killing forms and a Gallot-Meyer theorem for foliations[END_REF] and by using the fact that

d M (δ M ϕ) = d b (δ b ϕ) = λ 1,p ϕ = σ n+2-p (σ p+1 -g(S(ξ), ξ))ϕ, we deduce that    S [p-1] (ν φ) = (σ n+1 -σ n+2-p )ν φ S [p] (ϕ) = (σ p+1 -g(S(ξ), ξ))ϕ. (9)
Now, in order to prove that the manifold N is isometric to the quotient Γ \ R × B , we need first to establish a series of lemmas:

Lemma 4.1 If the equality is realized in (1), then S(ξ) = 0.

Proof. Using Equation ( 7), we deduce that the form ν φ is basic (recall here that δ M ϕ = δ b ϕ).

Hence by applying the first equation in [START_REF] Li | A sharp comparison theorem for compact manifolds with mean convex boundary[END_REF] to the vector fields ξ and

X 1 , • • • , X p-2 ∈ Γ(Q)
, we find that S(ξ) (ν φ) = 0. On the other hand, since the O'Neill tensor vanishes, then ∇ M ξ ϕ = ∇ ξ ϕ which is equal to zero, because the form ϕ is basic. Here, we recall that ∇ is the extension of the transversal Levi-Civita connection to basic forms. Finally, by taking X = ξ in the first equation of ( 8), we find that S(ξ) ∧ (ν φ) = 0. Mainly, that means S(ξ) = 0. We mention here that ν φ cannot vanish, since this would imply that ∇ M X ϕ = 0 for all X ∈ Γ(T M ) which would give λ 1,p = 0 (recall that λ 1,p is the first non-zero eigenvalue).

In the sequel, we aim to prove that the principal curvatures of S are constant and are all equal to a number η, along transversal principal directions. The proof of this statement is a technical computation and will be splitted into several lemmas (see Lemmas 4.2, 4.4 and 4.5). In the sequel, {f i } i=1,••• ,n+1 will denote an orthonormal frame of Γ(T M ).

(∇ M f i S) [p] ϕ, f i ∧ (ν φ) = ((σ p+1 -σ n+1 + σ n+2-p )σ n+2-p -|S| 2 )|ν φ| 2 + n+1 i=1 f i (ν φ), S 2 (f i ) (ν φ) . (10)
Proof. By differentiating the second equation in (9) along any vector field X ∈ Γ(T M ), we get after using (18)

S [p] (SX ∧ (ν φ)) + (∇ M X S) [p] ϕ = σ p+1 SX ∧ (ν φ).
Here we also used the first equation in [START_REF] Jung | Transversal conformal Killing forms and a Gallot-Meyer theorem for foliations[END_REF]. Setting X = f i and taking the scalar product of the last equality with f i ∧ (ν φ), we obtain after tracing and using (19) that,

n+1 i=1 (∇ M f i S) [p] ϕ, f i ∧ (ν φ) = σ p+1 n+1 i=1 S(f i ) ∧ (ν φ), f i ∧ (ν φ) - n+1 i=1 S 2 (f i ) ∧ (ν φ), f i ∧ (ν φ) - n+1 i=1 S(f i ) ∧ S [p-1] (ν φ), f i ∧ (ν φ) .
Then with the help of ( 9), the last equality reduces to

n+1 i=1 (∇ M f i S) [p] ϕ, f i ∧ (ν φ) = (σ p+1 -σ n+1 + σ n+2-p ) n+1 i=1 S(f i ) ∧ (ν φ), f i ∧ (ν φ) - n+1 i=1 S 2 (f i ) ∧ (ν φ), f i ∧ (ν φ) . (11)
In order to finish the proof, it is sufficient to calculate the two sums in the r.h.s. of [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF]. In fact, the first sum is equal to

n+1 i=1 S(f i ) ∧ (ν φ), f i ∧ (ν φ) = σ n+1 |ν φ| 2 - n+1 i=1 f i (ν φ), S(f i ) (ν φ) (20), (9)
= σ n+2-p |ν φ| 2 , while the second one is

n+1 i=1 S 2 (f i ) ∧ (ν φ), f i ∧ (ν φ) = |S| 2 |ν φ| 2 - n+1 i=1 S 2 (f i ) (ν φ), f i (ν φ) .
The substitution into [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF] gives the desired result.

In the following lemma, we will expand the curvature operator of the manifold N. Indeed, Lemma 4.3 We have the splitting

W N p (ν ∧ (ν φ)), ϕ = I + J + K, (12)
where I, J and K are given by

I = n+1 i=1 R N (ν, f i )ν φ, f i ϕ J = (-1) p(p-1) 2 n+1 i=1 R N (ν, f i , f i , (ν φ) ϕ) K = - n+1 i,j=1 R N (f i , f j )ν ∧ (f i (ν φ)) , f j ϕ .
Moreover, we have that I = K. When the equality is realized in (1), we get

I = -(-1) p(p-1) 2
((ν φ) ϕ)(σ n+1 ) + (σ n+1 -σ n+2-p )σ p+1 |ϕ| 2 -σ 2 p+1 |ϕ| 2 + n+1 i=1 f i ϕ, S 2 (f i ) ϕ . ( 13
)
Proof. Recall that the curvature W N p is given by the expression W N p := i,j e * j ∧ e i R N (e i , e j ) where

{e i } i=1,••• ,n+2 is any orthonormal frame of T N . At a point x ∈ M , we choose the orthonormal frame {f i , ν} i=1,••• ,n+1 of T x N to compute W N p (ν∧(ν φ)), ϕ = n+1 i=1 ν R N (ν, f i )(ν ∧ (ν φ)) , f i ϕ + n+1 i,j=1 f i R N (f i , f j )(ν ∧ (ν φ)) , f j ϕ = n+1 i=1 R N (ν, f i )ν φ, f i ϕ + n+1 i,j=1 R N (ν, f i )f i , f j ν φ, f j ϕ - n+1 i,j=1 R N (f i , f j )ν ∧(f i (ν φ)) , f j ϕ . = n+1 i=1 R N (ν, f i )ν φ, f i ϕ + (-1) p-1 n+1 i,j=1 R N (ν, f i )f i , f j (ν φ) ∧ f j , ϕ - n+1 i,j=1 R N (f i , f j )ν ∧ (f i (ν φ)) , f j ϕ .
Then using (21), we deduce [START_REF] Richardson | The basic Laplacian of a Riemannian foliation[END_REF]. To prove that I = K, we first remark that the term -K is equal to the following:

i 1 <•••<i p-1 k=1••• ,p-1 i,j=1,••• ,n+1 (-1) k+1 (ν φ) i 1 ,••• ,i p-1 δ ii k R N (f i , f j )ν ∧ f i 1 ∧ • • • fi k ∧ • • • ∧ f i p-1 , f j ϕ = i 1 <•••<i p-1 k=1••• ,p-1 j=1,••• ,n+1 (ν φ) i 1 ,••• ,i p-1 f i 1 ∧ • • • ∧ R N (f i k , f j )ν ∧ • • • ∧ f i p-1 , f j ϕ (2) = i 1 <•••<i p-1 k=1••• ,p-1 j=1,••• ,n+1 (ν φ) i 1 ,••• ,i p-1 f i 1 ∧ • • • ∧ (∇ M f j S)(f i k ) ∧ • • • ∧ f i p-1 , f j ϕ = n+1 j=1 ϕ, f j ∧ (∇ M f j S) [p-1] (ν φ) . (14)
Hence, using the second equation in ( 14), I -K is equal to

i 1 <•••<i p-1 k=1••• ,p-1 i=1,••• ,n+1 (ν φ) i 1 ,••• ,i p-1 { f i 1 ∧ • • • ∧ R N (ν, f i )f i k ∧ • • • ∧ f i p-1 , f i ϕ + f i 1 ∧ • • • ∧ R N (f i k , f i )ν ∧ • • • ∧ f i p-1 , f i ϕ } = i 1 <•••<i p-1 k=1,••• ,p-1 i,l=1,••• ,n+1 (ν φ) i 1 ,••• ,i p-1 {R N (f i k , f l , ν, f i ) f i 1 ∧ • • • ∧ f l ∧ • • • ∧ f i p-1 , f i ϕ +R N (f i k , f i , ν, f l ) f i 1 ∧ • • • ∧ f l ∧ • • • ∧ f i p-1 , f i ϕ },
which is zero when we interchange the role of the indices i and l in the first summation. When the equality is realized in (1), and since 1] (ν φ) , we get

I = K = - n+1 i=1 f i ϕ, (∇ M f i S) [p-
I (18) = - n+1 i=1 f i ϕ, ∇ M f i (S [p-1] (ν φ)) + n+1 i=1 f i ϕ, S [p-1] (∇ M f i (ν φ)) (8),(9)
= -

n+1 i=1 f i (σ n+1 ) f i ϕ, ν φ + (σ n+1 -σ n+2-p ) n+1 i=1 f i ϕ, S(f i ) ϕ - n+1 i=1 f i ϕ, S [p-1] (S(f i ) ϕ) ,
which gives the result, using Equations ( 21),(20),( 23) and ( 9).

In the following, we will compute the l.h.s. of Equation [START_REF] O'neill | The fundamental equations of a submersion[END_REF] in terms of the curvature operator W N p . Indeed, we have Lemma 4.4 If the equality is realized in (1), we have

n+1 i=1 (∇ M f i S) [p] ϕ, f i ∧ (ν φ) = -W N p (ν ∧ (ν φ)), ϕ + n+1 i=1 f i ϕ, S 2 (f i ) ϕ +(σ n+1 -σ n+2-p -σ p+1 )σ p+1 |ϕ| 2 . ( 15
)
Proof. Using the symmetry property of the tensor ∇ M S and Equation ( 19), the l.h.s. of Equation ( 15) is equal to

n+1 i=1 ϕ, (∇ M f i S)(f i ) ∧ (ν φ) + ϕ, f i ∧ (∇ M f i S) [p-1] (ν φ) (21)
= (-1)

p(p-1) 2 n+1 i=1 (ν φ) ϕ, (∇ M f i S)(f i ) + n+1 i=1 ϕ, f i ∧ (∇ M f i S) [p-1] (ν φ) .
Therefore, from Equation ( 2) and again from the symmetry of ∇ M S, the above expression reduces to (-1)

p(p-1) 2 n+1 i=1 (∇ M (ν φ) ϕ S)(f i ), f i + (-1) p(p-1) 2 R N ((ν φ) ϕ, f i , ν, f i ) + n+1 i=1 ϕ, f i ∧ (∇ M f i S) [p-1] (ν φ) = (-1) p(p-1) 2
((ν φ) ϕ)(σ n+1 ) -(-1) p(p-1) 2 n+1 i=1 R N (ν, f i , f i , (ν φ) ϕ) + n+1 i=1 ϕ, f i ∧ (∇ M f i S) [p-1] (ν φ) . (12),(14) = (-1) p(p-1) 2 ((ν φ
) ϕ)(σ n+1 ) -W N p (ν ∧ (ν φ)), ϕ + I + K -K. (16)
Substituting Equation ( 13) into Equation ( 16), we finally get the result.

In the next lemma, we will compare the sign of the l.h.s. of Equation [START_REF] O'neill | The fundamental equations of a submersion[END_REF] which is given by [START_REF] Semmelmann | Conformal Killing forms on Riemannian manifolds[END_REF] to the sign of the r.h.s. Under a curvature assumption, we will find that they are of opposite signs. In particular, this will mean that all principal curvatures along transversal directions are equal. More precisely, we have Lemma 4.5 If the equality in (1) is realized and if moreover σ 1 (M ) ≥ 0, then S(X) = ηX for all X orthogonal to ξ.

Proof. We will show that the l.h.s. of Equation ( 10) is nonnegative while the r.h.s. is nonpositive which means that both terms vanish. We first begin to check the l.h.s. which is given by Equation ( 15) in Lemma 4.4. Indeed, the eigenform φ is parallel thus the term W N p φ, φ vanishes. Therefore by writing φ = ϕ+ν∧(ν φ) at any point of the boundary and using the fact that W N p is nonnegative, we deduce that the term W N p (ν ∧ (ν φ)), ϕ is nonpositive. On the other hand, the tensor S has 0 as an eigenvalue (recall that S(ξ) = 0) and that σ 1 (M ) ≥ 0, therefore all the η i 's are greater than

0 for i = 2, • • • , n + 1. Hence n+1 i=1 f i ϕ, S 2 (f i ) ϕ ≥ η 2 n+1 i=1 f i ϕ, S(f i ) ϕ (20),(9) = η 2 σ p+1 |ϕ| 2 .
Thus, the l.h.s. of Equation ( 10) is bounded from below by

(σ n+1 -σ n+2-p -σ p+1 + η 2 )σ p+1 |ϕ| 2 = ((η n+3-p -η 3 ) + • • • + (η n+1 -η p+1 ))σ p+1 |ϕ| 2 ≥ 0,
since the sequence η i is increasing. We can easily see that all inequalities are sharp when all the η i s (i = 2, • • • , n + 1) are equal. Concerning the r.h.s. of Equation [START_REF] O'neill | The fundamental equations of a submersion[END_REF], recall that it is given by

((σ p+1 -σ n+1 + σ n+2-p )σ n+2-p -|S| 2 )|ν φ| 2 + n+1 i=1 f i (ν φ), S 2 (f i ) (ν φ) . (17)
In the sequel, we will take the vectors {f i } i=1,••• ,n+1 as the principal directions associated with the principal curvatures η i of the tensor S. We first have

n+1 i=1 f i (ν φ), S 2 (f i ) (ν φ) = n+1 i=1 η i f i (ν φ), S(f i ) (ν φ) ≤ η n+1 n+1 i=1 f i (ν φ), S(f i ) (ν φ) (20),(9) = η n+1 (σ n+1 -σ n+2-p )|ν φ| 2 .
Hence, ( 17) can be bounded from above by A|ν φ| 2 , where A is given by

A = (σ p -σ n+1 + σ n+2-p )σ n+2-p + η p+1 σ n+2-p -η 2 2 -• • • -η 2 n + η n+1 (η n+3-p + • • • + η n ).
Let us prove that A is nonpositive, which implies in particular that ( 17) is nonpositive. We write

A = (σ p -σ n+1 + σ n+2-p )σ p+1 + (σ p -σ n+1 + σ n+2-p )(η p+2 + • • • + η n+2-p ) +η p+1 σ p + η p+1 (η p+1 + • • • + η n+2-p ) -η 2 2 -• • • -η 2 n + η n+1 (η n+3-p + • • • + η n ) = σ p σ p+1 -(σ n+1 -σ n+2-p )σ p+1 + (σ p -σ n+1 + σ n+2-p )(η p+2 + • • • + η n+2-p ) +η p+1 σ p + B -η 2 2 -• • • -η 2 p -η 2 n+3-p -• • • -η 2 n + η n+1 (η n+3-p + • • • + η n )
where B is given by

B = η p+2 (η p+1 -η p+2 ) + η p+3 (η p+1 -η p+3 ) + • • • + η n+2-p (η p+1 -η n+2-p ).
Clearly B is nonpositive, since 1 ≤ p ≤ n 2 . On the other hand, using that S(ξ) = 0 we have

S [p] (ξ ∧ (ν φ)) (19) = ξ ∧ S [p-1] (ν φ) (9)
= (σ n+1 -σ n+2-p )(ξ ∧ (ν φ)), [2] (just apply S [2] to the eigenform (f n+3-p ∧ • • • ∧ f n ) ϕ by using the formula ( 22)). This fact combined with η i ≥ η p+2 for i

then σ p + σ n+2-p ≤ σ n+1 . Therefore A ≤ σ 2 p + 2η p+1 σ p -(σ n+1 -σ n+2-p )σ p+1 -η 2 2 -• • • -η 2 p -η 2 n+3-p -• • • -η 2 n + η n+1 (η n+3-p + • • • + η n ) = 2 2≤i<j≤p η i η j + 2η p+1 σ p -η n+3-p (η n+3-p + σ p+1 ) -• • • -η n (η n + σ p+1 ) -η n+1 (σ p+1 -η n+3-p -• • • -η n ). It is clear that η n+3-p + • • • + η n ≤ σ p+1 , as σ p+1 -(η n+3-p + • • • + η n ) is an eigenvalue of S
= n + 3 -p, • • • , n + 1, gives that A ≤ 2 2≤i<j≤p η i η j + 2η p+1 σ p -η p+2 (p -1)σ p+1 ≤ 0.
This last inequality is true because the number of positive terms is equal to the number of negative terms which is p(p -1). Now we are able to prove Theorem 1.2.

Proof of Theorem 1.2. The proof follows exactly the same lines as in [4, Thm. 1.2] (see also [START_REF] Chami | Rigidity results for spin manifolds with foliated boundary[END_REF] for more details) that we briefly explain the main idea. We first show that the vector field ξ defining the flow can be extended to a unique parallel vector field ξ on N which is orthogonal to ν [START_REF] Chami | Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary[END_REF]. The proof mainly relies on the use of the Reilly formula applied to the solution ξ of the boundary value problem [START_REF] Chami | Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary[END_REF]. Second, we consider a connected integral submanifold N 1 of the bundle (R ξ) ⊥ , where the orthogonal complement is taken in N . The manifold N 1 is complete with totally umbilical boundary and the Ricci tensor of ∂N 1 is bounded from below by the constant (n -1)η 2 . That is, the manifold ∂N 1 is compact as a consequence of Myers's theorem and from the main result in [9, Thm. 1.1], we deduce that N 1 is compact.

On the other hand, we have, from [START_REF] Jung | Transversal conformal Killing forms and a Gallot-Meyer theorem for foliations[END_REF], ϕ = -1 pη d M (ν φ) which means that it is d M -exact and thus d ∂N 1 -exact, since ∂N 1 is totally geodesic in M and both ϕ and ν φ are basic. Moreover, the basic form ϕ is an eigenform of the Laplacian on ∂N 1 , that is ∆ ∂N 1 ϕ = ∆ b ϕ = λ 1,p ϕ. Therefore, if we denote by λ ∂N 1 1,p the first eigenvalue of ∆ ∂N 1 restricted to exact p-forms on N 1 and by σp the p-curvatures of ∂N 1 into the compact manifold N 1 , we get from the main estimate in [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF]Thm. 5

] that p(n + 1 -p)η 2 = σp σn+1-p ≤ λ ∂N 1 1,p ≤ λ 1,p = σ p+1 σ n+2-p = p(n + 1 -p)η 2 .
Hence the equality is attained in the estimate of S. Raulot and A. Savo [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF]Thm. 5] and therefore N 1 is isometric to the Euclidean closed ball B . Finally, by the de Rham theorem, the manifold N is isometric to R × B and N is the quotient of the Riemannian product R × B by its fundamental group. Since π 1 (N ) embeds into π 1 (M ), the manifold N is then isometric to Γ \ R × B , for some fixed-point-free cocompact discrete subgroup Γ ⊂ R × SO n+1 .

Rigidity results on manifolds with foliated boundary

Our objective, in this section, is to derive rigidity results on manifolds with foliated boundary. These results generalize the ones in [START_REF] Chami | Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary[END_REF]Sect. 5]. For this end, we recall that a basic special Killing p-form ω is a basic co-closed (with respect to the basic codifferential δ b ) form satisfying for all X ∈ Γ(Q) the relations [START_REF] Schwarz | Hodge decomposition-A method for solving boundary value problems[END_REF][START_REF] Tachibana | On a Riemannian space admitting more than one Sasakian structures[END_REF][START_REF] Jung | Transversal conformal Killing forms and a Gallot-Meyer theorem for foliations[END_REF]]

∇ X ω = 1 p + 1 X d b ω and ∇ X d b ω = -c(p + 1)X ∧ ω,
where ∇ is the transversal Levi-Civita connection and c is a non-negative constant. In general, one can prove that a basic special Killing p-form is a co-closed eigenform of the basic Laplacian corresponding to the eigenvalue c(p + 1)(n -p) where n is the rank of Q.

In the following, we will consider a compact manifold N whose boundary carries a basic special Killing p-form. Due to the equality case of our main estimate, we will be able to characterize the boundary as the product S 1 × S n . We first prove the following result: Next, we define the interior product of an s-form with a p-form ϕ, as follows:

((X 1 ∧ • • • ∧ X s ) ϕ)(Y 1 , • • • , Y p-s ) = ϕ(X s , • • • , X 1 , Y 1 , • • • , Y p-s ).
Therefore, for any (p -s)-form ψ, one has

(X 1 ∧ • • • ∧ X s ) ϕ, ψ = (-1) s(s-1) 2 ϕ, X 1 ∧ • • • ∧ X s ∧ ψ . (21)
Finally,

S [p-s] ((X 1 ∧ • • • ∧ X s ) ϕ) = (X 1 ∧ • • • ∧ X s ) S [p] (ϕ) -(S [s] (X 1 ∧ • • • ∧ X s )) ϕ. (22)
In particular, this gives for s = 1

S [p-1] (X ϕ) = X S [p] (ϕ) -S(X) ϕ. (23)

1 . 1 ]

 11 by the additional term 4[ n 2 ] sup M |Ω| 2 that results

  For a compact manifold M , we consider the L 2 -adjoint of d b , denoted by δ b , and define the basic Laplacian as ∆ b = d b δ b + δ b d b .

Lemma 4 . 2

 42 If the equality is realized in (1), then n+1 i=1

Corollary 5 . 1

 51 Let N be an (n + 2)-dimensional compact manifold with non-negative curvature operator. Assume that the boundary M carries a minimal Riemannian flow such that (n + 1p) sup M g(S(ξ), ξ) + 4c 2 [ n 2 ] ≤ 0 and also admits a basic special Killing (n -p)-form for some2 ≤ p ≤ n 2 . If the inequality σ p+1 (M ) ≥ p holds, the manifold N is isometric to Γ \ R × B .Proof. Let ϕ be a basic special Killing (n -p)-form on M . Then * b ϕ is a basic closed p-eigenform for the basic Laplacian, that is ∆ b ( * b ϕ) = p(n + 1 -p)( * b ϕ). Here, we used the minimality of the flow to say that the basic Hodge operator " * b " commutes with the basic Laplacian. Hence

Acknowledgment: The first two named authors were supported by a fund from the Lebanese University. The second named author would like to thank the Alexander von Humboldt Foundation for its support.

λ 1,p ≤ p(n + 1 -p) [START_REF] Ph | Foliations on Riemannian manifolds[END_REF]. To get the upper bound, we will use the estimate in Theorem 1.1. First, we have σ n+1-p ≥ σ p+1 > 0 as 2 ≤ p ≤ n 2 . On the other hand, by considering the functions θ i = σ i+1 -η 1 for all i = 1, • • • , n, and using the estimate

we finish the proof because η 1 ≤ g(S(ξ), ξ) ≤ 0.

Using this last result, we can prove Corollary 1.3 as in [START_REF] Chami | Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary[END_REF]Cor. 1.3]. Also a similar result holds as in [START_REF] Chami | Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary[END_REF]Cor. 5.2] Corollary 5.2 Let N be an (n + 2)-dimensional compact manifold with non-negative curvature operator. Assume that M = S 1 × S n with n ≥ 3, the sectional curvature K N of N vanishes on M, the mean curvature H > 0 and

Finally, the analogue result holds as in [START_REF] Chami | Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary[END_REF]Cor. 5.3] Corollary 5.3 Let N be an (n + 2)-dimensional compact manifold with non-negative curvature operator. Assume that the boundary M carries a minimal Riemannian flow such that (n + 1p) M g(S(ξ), ξ) + 4c 2 [ n 2 ] ≥ 0 and also admits a basic special Killing (n -p)-form for some

Proof. We follow the same proof as in Corollary 5.1. We just remark that

which finishes the proof of the corollary.

Appendix

In this Appendix, we will state some technical formulas that we use in our computations. We omit the proofs of these formulas and leave them to the reader. For any X ∈ Γ(T M ), we have

Also, for any p-form ϕ and X ∈ Γ(T M ),

For any orthonormal frame {f i } i=1,••• ,n+1 of T M , we have


Raphael Viera
email: raphael.viera@emse.fr

Jean-Max Dutertre
email: dutertre@emse.fr

Mathieu Dumont
email: mathieu.dumont@cea.fr

Pierre-Alain Moëllic
email: pierre-alain.moellic@cea.fr

Permanent Laser Fault Injection into the Flash Memory of a Microcontroller

The Flash memory of a Microcontroller Unit (MCU) is an important part of its attack surface as it contains its firmware and its security related data (e.g. passwords and cryptographic keys). Recent research works report the use of Laser Fault Injections (LFI) to corrupt the firmware at run time by targeting the Flash memory during its read operations (data reads from Flash were also faulted). These faults, induced on a single bit and following a bit-set fault model, were nonpermanent: the data stored in Flash stayed unaltered while only their read copies were corrupted. We report an extension of this fault model on the Flash memory of a 32-bit MCU. Using LFI, we were able to induce permanent faults into its Flash. Single bit faults, that followed a bit-reset fault model, were induced during the Flash write operations. As a proof of concept, we describe how we were able to iteratively set to zero all the bits of a 32bit password using a laser pulse with relatively undemanding settings (15 µm beam diameter, and 3 µs pulse duration).

I. INTRODUCTION

In the field of secure circuits characterization, physical attacks constitute a major threat for security engineers and semiconductors manufacturers. When an attacker has a physical access to a MCU, he can exploit hardware-based vulnerabilities in order to retrieve secret information.

Sensitive data, such as encryption keys or passwords, are usually stored in a non-volatile memory (NVM): the Flash memory, which is frequently embedded on MCUs. Fault injection analysis (FIA) have become a common way to defeat the security mechanisms of these devices [START_REF] Barenghi | Fault injection attacks on cryptographic devices: Theory, practice, and countermeasures[END_REF].

Previous FIA related works demonstrate attacks in a Flash memory by means of LFI (Laser Fault Injection). The attacks reported in [START_REF] Skorobogatov | Optical fault masking attacks[END_REF], [START_REF] Cai | Optical fault injection attacks for flash memory of smartcards[END_REF] aimed at preventing changes of the memory content by disrupting its normal operations. Recent studies [START_REF] Colombier | Laser-induced Single-bit Faults in Flash Memory: Instructions Corruption on a 32-bit Microcontroller[END_REF]- [START_REF] Garb | Temporary laser fault injection into flash memory: Calibration, enhanced attacks, and countermeasures[END_REF] demonstrated on 32-bit MCUs embedding a NOR Flash that LFI induces non permanent faults when instructions and data are read from the Flash. The stored data were left unmodified while the read data were faulted one bit at a time following a bit-set fault model (i.e. a 0 is turned into a 1). Changing the location of the laser beam made it possible to choose what bit of a 32-bit word was faulted with a 100% success rate.

This paper reports an improvement of the previous threats as we were able to induce permanent faults into a MCU's Flash memory using LFI during write operations. For the same locations used to induce faults in a Flash during read operations, we induced single bit bit-reset faults. Then to validate this fault model, we set to zero a password stored in the target's Flash memory.

This article is organized as follows. Section II provides the background information on NOR Flash memories arschitecture and LFI. Section III describes our laser injection setup and the used MCU target. The obtained experimental results are given in section IV. Finally a conclusion is drawn in section V.

II. BACKGROUND

This section describes the basic architecture and operating modes of the MCU's embedded Flash memory: a NOR Flash for the circuit we considered. We also describe the effect of LFI on a Flash memory.

A. NOR Flash Architecture

NOR Flash memory was born in mid 80's as an EPROM replacement [START_REF] Campardo | VLSI-Design of Non-Volatile Memories[END_REF]. The advanced architecture of NOR Flash memory is effectively conceived to meet the requirements of MCU by optimizing the trade-off between speed and power consumption. It also gives the possibility of using a single chip to store both user code and user data.

All the features of the NOR Flash are inherently related to the memory cell concept and its memory array organization as depicted in Fig. 1. Each transistor is called a cell and it is made of a stacked-double-poly floating-gate MOS device. The memory cells are arranged in a NOR type array organization, which means that all the cells are parallel connected with a common ground node and the bit lines are directly connected to the drains of memory cells. Read: to read a value from the Flash memory at the application level, it is sufficient to create a pointer to the specific address and read its content as a common memory space. At the electrical level, the threshold voltage of a cell is measured by a sense amplifier which in turn determines if its content is equivalent to a logic 0 (programmed) or logic 1 (erased) [START_REF] Campardo | VLSI-Design of Non-Volatile Memories[END_REF].

Program (Write): the embedded Flash memory can be programmed at run time using in-application programming (download programming data into memory while the application is running). In the case of our target (see III-B), only 16 bits (from a 32-bit word) can be programmed at a time. At the electrical level, each cell can be programmed by using channel-hot-electron injection [START_REF] Campardo | VLSI-Design of Non-Volatile Memories[END_REF]. As a result, electrons are pushed into the floating gate of a cell. At logical level, a programmed cell stores a logical 0 (it is not possible to write a 1, which is done through erasing).

Erase: at the physical level, the erase operation is done using the Fowler-Nordheim tunneling effect [START_REF] Campardo | VLSI-Design of Non-Volatile Memories[END_REF]. As a result, the electrons stored into a cell's floating gate are removed. At logical level, an erased cell stores a logical 1. In the case of our target (see III-B), erasing is done on a whole memory page: 1 kB of data.

Hence, writing data (made of both 0s and 1s) in a Flash Memory is made in several steps: 1 st a given section of the memory is erased, 2 nd 0s are written in the relevant places.

C. Flash Memory Laser Fault Model

When a laser beam passes through the silicon substrate of a target, it creates electron-hole pairs along the path of the laser. If these induced charge carriers reach the strong electric field found in the vicinity of reverse biased PN junctions, the electrical field puts these charges into motion and a transient current flows. [START_REF] Colombier | Laser-induced Single-bit Faults in Flash Memory: Instructions Corruption on a 32-bit Microcontroller[END_REF] and [START_REF] Menu | Single-bit Laser Fault Model in NOR Flash Memories: Analysis and Exploitation[END_REF] describe how such a transient current may discharge the bitline of a Flash memory cell during a read operation resulting in a bit-set transient fault. As LFI is local, it makes it possible to fault a single bit and to choose its index while reading a 32-bit word with 100% success rate.

III. EXPERIMENTAL METHOD

A. Laser setup for the experimental fault injection experimental results reported in the next sections were obtained through the target backside using a laser source with a 1,064 nm wavelength (near infrared range). This source was used to generate laser pulses of a duration equal to 3 µs and a power of 200 mW. By using an objective with x5 magnification, the obtained laser spot diameter is 15 µm.

B. Target board and microcontroller

The target MCU we used for our experiments embeds an ARM Cortex-M3 core with 128 kB of Flash memory. It is manufactured in the CMOS 90 nm technology node. Its clock was set to 7.4 MHz. An infrared picture of the target microcontroller is shown in Fig. 3. The chip has a dimension of 3 mm x 2.5 mm. The main parts of the chip are outlined including the Flash memory which constitutes the target of this experiment. Since LFI requires the surface of the chip's die being visible, the microcontroller packaging was milled away with engraving tools. The chip was then placed into a test board compatible with the ChipWhisperer platform [START_REF] O'flynn | ChipWhisperer: An open-source platform for hardware embedded security research[END_REF]. The board was mounted on a motorized XYZ-stage and a CCD camera mounted on optical objectives (Fig. 4).

Target's Flash memory organization: it has a main memory block containing 128 pages of 1 kB and an information block as shown in Table I (the corresponding base addresses in the memory map are given). The memory is organized as 32-bit wide memory words that can be used for storing both code and data.

C. Embedded Software and Aim of Experiment

Our work aimed at demonstrating the ability of LFI to inject permanent faults at write time into a Flash memory. To that end, we wrote a dedicated test code described by the pseudocode in listing 1. The same code was used by the experiments reported in sections IV-A and IV-B. The code begins with the main() function, it waits for an UART message that will call the program_memory() function. Using FPEC rules, this function writes into the Flash memory, at address 0x0801FFF0, the value 0xFFFFFFFF for section IV-A or the original password (0xABCD2021) for section IV-B. If is_init=false, the program will read the content of the memory address and rewrite the same content again (used by the experiment of section IV-B), otherwise it will only write the original value into memory. Each writing operation calls the FPEC_write() function which sets a synchronization trigger for the laser source. The laser is then shot after a programmed delay. As reported in the next section IV, at each iteration of the writing operation, a specific bit in memory can be reset from logic '1' to logic '0'.

IV. RESULTS

A. LFI into a Flash memory during write operation

Before running the experiment to overwrite the original and unknown password stored in the Flash memory, we studied which parts of the Flash memory are LFI sensitive. For this experiment, the value 0xFFFFFFFF was written at the memory address 0x0801FFF0.

As highlighted in listing 1, each write operation provides a trigger signal (it is drawn in black in Fig. 5) that outlines the duration of this operation. An electronic synchronization board is then used to deliver a shot signal to the laser source, denoted Pulse Picker in Fig. 5 (blue) after a programmable delay. An image of the actual laser pulse is also depicted in red; its pulse width was set to 3 µs. In order to write 32 bits into the memory, FPEC determines that two 16-bit write operations are necessary (as shown in Fig. 5). A write operation lasts 56 µs, setting the delay (measured from the rising edge of the trigger signal to the shot signal) at 44 µs, we were able to induce permanent bit-reset faults into one bit at the written address of the Flash memory. and ∆Y=100 µm displacement steps. Moving the laser spot along the X-axis made it possible to fault independently every bit of a 32-bit word; locations where faults were induced are marked in red: for Y=100 µm the 32 bit locations are clearly visible. While changing the laser spot Y coordinates had no effect on the faulted-bit. These patterns of LFI sensitivity are the same that were observed by [START_REF] Colombier | Laser-induced Single-bit Faults in Flash Memory: Instructions Corruption on a 32-bit Microcontroller[END_REF]- [START_REF] Garb | Temporary laser fault injection into flash memory: Calibration, enhanced attacks, and countermeasures[END_REF] when inducing faults during a read operation; except that they induced bit-set faults. This experiment assessed that single bit-reset faults can be induced by laser during a write operation in a predictable and repeatable manner using a large 15 µm laser spot diameter.

B. Password zeroing

As a proof of concept, we then applied this new fault model to iteratively set to zero a 32-bit password stored in Flash. We used the code in Listing 1 with the Y-axis position set to 100 µm, while the X-axis was swept from 0 µm to 1400 µm with a ∆X=5 µm step. For this experiment we assumed that the password was unknown (though its value was 0xABCD2021), and that the only available information was its address in memory (0x0801FFF0). The experiment aim was to reset the password to 0x00000000 (a known value). In a conservative approach, we used 280 steps ( 1400 µm 5 µm ) along the X axis to iterate on the 32 bits of the password. As a result, all the 1s in the password were reset to zero (the bits already at 0 were not modified). This process is illustrated in Table II which depicts actual data from the experiments and highlights (in red) the induced bit-resets. Therefore the unknown password was replaced by a known all-0s one that can be used to give access to secure information saved in the Flash memory. By performing this experiment several times, we could observe the same result each time, thus giving a 100% success rate.

V. CONCLUSIONS AND DISCUSSION

Our experiments assess that LFI can be used during the write operations of a Flash memory: single bit-reset faults 1010 1011 1000 0000 0000 0000 0000 0000 23 0xAB000000 1010 1011 0000 0000 0000 0000 0000 0000 24 0xAA000000 1010 1010 0000 0000 0000 0000 0000 0000 25 0xA8000000 1010 1000 0000 0000 0000 0000 0000 0000 27 0xA0000000 1010 0000 0000 0000 0000 0000 0000 0000 29 0x80000000 1000 0000 0000 0000 0000 0000 0000 0000 31 0x00000000 0000 0000 0000 0000 0000 0000 0000 0000 are induced repeatably when targeting the Flash array. This is an extension of the previous state-of-the-art [START_REF] Colombier | Laser-induced Single-bit Faults in Flash Memory: Instructions Corruption on a 32-bit Microcontroller[END_REF]- [START_REF] Garb | Temporary laser fault injection into flash memory: Calibration, enhanced attacks, and countermeasures[END_REF] that reported laser-induced single bit-sets during read operations.

As an illustration we reset a 32-bit password stored in the Flash memory. These results were obtained using a laser beam diameter of 15 µm and a pulse duration of 3 µs, which are relatively undemanding settings (i.e. that an attacker can access more easily). This work is a first proof of feasibility, its relevance may be increased by devising a more practical attack scenario. Though we used a trigger signal to synchronize the laser shots with the target activity, it can be replaced by observing the electrical activity of the target as a write operation involves the activation of the MCU charge pump to produce the required high voltage.

At this point, the underlying physical phenomenon explaining the bit-reset fault model is still to be identified (that of the LFI-induced bit-set during read operations is given in [START_REF] Colombier | Laser-induced Single-bit Faults in Flash Memory: Instructions Corruption on a 32-bit Microcontroller[END_REF]). Testing the use of multi-spots LFI is another prospective work that may increase further the associated threat.

Figure 1 :

 1 Figure 1: NOR Flash memory architecture.

Figure 2 :

 2 Figure 2: Operating modes of the Flash memory: Program (Write), Erase, and Read.

Figure 3 :

 3 Figure 3: Infrared picture of the targeted microcontroller.

Figure 4 :

 4 Figure 4: Objective lenses above the targeted 32-bit MCU mounted on the Chipwhisperer platform.

Listing 1 :

 1 Program used to perform the experiments reported in sections IV-A and IV-B

Figure 5 :

 5 Figure 5: Voltage waveforms sampled during Flash writing operations: a trigger signal (in black) outlined every write operation (16-bit). It is used to synchronize the laser shots with the write operations (the laser command is drawn in blue and the laser shot itself in red).

Fig. 6 Figure 6 :

 66 Fig.6reports the LFI sensitivity scan we carried out on the Flash array surface (targeting its memory cells) with ∆X=5 µm

Table I :

 I MCU's embedded Flash module organization.

	Block	Name	Base Address	Size
		Page 0	0x0800 0000 -0x0800 03FF	1 kB
	Main mem.	...	...	...
		Page 127	0x0801 FC00 -0x0801 FFFF	1 kB
	Information	System mem. Option Bytes	0x1FFF F000 -0x1FFF F7FF 0x1FFF F800 -0x1FFF F80F	2 kB 16

Table II :

 II Iterative zeroing of the password (faults in red).

	Bit #	Value (Hex)	Value (Bin)
	-	0xABCD2021	1010 1011 1100 1101 0010 0000 0010 0001
	0	0xABCD2020	1010 1011 1100 1101 0010 0000 0010 0000
	5	0xABCD2000	1010 1011 1100 1101 0010 0000 0000 0000
	14	0xABCD0000	1010 1011 1100 1101 0000 0000 0000 0000
	16	0xABCC0000	1010 1011 1100 1100 0000 0000 0000 0000
	18	0xABC80000	1010 1011 1100 1000 0000 0000 0000 0000
	19	0xABC00000	1010 1011 1100 0000 0000 0000 0000 0000
	22	0xAB800000