Hypersurfaces d'un fibré vectoriel Riemannien à courbure de Gauss prescriteHypersurfaces of a Riemannian vector bundle with prescribed Gaussian curvature

Let M be a compact Riemannian manifold, E a Riemannian vector bundle on M and Σ the sphere subbundle of E. We look for embeddings of Σ into E admitting prescribed Gaussian curvatures of various types. To cite this article: A. Hanani, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 927–930.     

Asymptotic expansion of the Faber–Krahn profile of a compact Riemannian manifold

The aim of this Note is to give a proof of a well-known fact: an asymptotic expansion of the isoperimetric profile of a Riemannian manifold for small volumes gives an asymptotic expansion of the Faber–Krahn profile for this same Riemannian manifold. To cite this article: O. Druet, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Hypersurfaces compactes d'un fibré vectoriel riemannien à courbure moyenne prescriteCompact hypersurfaces of a Riemannian vector bundle with prescribed mean curvature

Let M be a compact Riemannian manifold, E a Riemannian vector bundle on M and Σ the sphere subbundle of E. We look for embeddings of Σ into E admitting prescribed mean curvatures of various types. To cite this article: P. Cherrier, A. Hanani, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 525–528.     

Minimal singular Riemannian foliations

We prove that a singular foliation on a compact manifold admitting an adapted Riemannian metric for which all leaves are minimal must be regular. To cite this article: V. Miquel, R.A. Wolak, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Differentiability of the isoperimetric profile and topology of analytic Riemannian manifolds

We show that smooth isoperimetric profiles are exceptional for real analytic Riemannian manifolds. For instance, under some extra assumptions, this can happen only on topological spheres. To cite this article: R. Grimaldi et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Homogeneous Einstein–Randers spaces of negative Ricci curvature

We prove that a homogeneous Einstein–Randers space with negative Ricci curvature must be Riemannian. To cite this article: S. Deng, Z. Hou, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Remarques sur les spineurs de Killing transversaux

We show necessary conditions for the existence of transversal Killing spinors on a spin manifold endowed with a Riemannian flow. To cite this article: N. Ginoux, G. Habib, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Complete intersections with metrics of positive scalar curvature

We give a complete list of complex projective complete intersections admitting Riemannian metrics of positive scalar curvature. To cite this article: F. Fang, P. Shao, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Stochastic calculus of variations and Harnack inequality on Riemannian path spacesCalcul de variations stochastiques et l'inéqualité de Harnack sur l'espace de chemins riemanniens

We describe the tangent space of Riemannian path space as a space of tangent processes localized on Brownian sheets; the bundle of adapted frames above a Riemannian path space and its structural equation are given. The stochastic calculus of variations allows us to derive Harnack–Bismut inequality for the Norris semigroup. To cite this article: A.-B. Cruzeiro, P. Malliavin, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 817–820.     

Vector fields dynamics as geodesic motion on Lie groups

We study to what extent vector fields on Lie groups may be considered as geodesic fields. For a given left invariant vector field on a Lie group, we prove there exists a Riemannian metric whose geodesics are its trajectories. When we consider left invariant metrics, differences between the Riemannian and the Lorentzian cases appear, coded by properties of the Lie algebra. To cite this article: G.T. Pripoae, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On self attracting/repelling diffusionsDiffusions auto attractives/repulsives

We present an almost sure ergodic theorem for a class of self-interacting diffusions on a compact Riemannian manifold. To cite this article: M. Benaim, O. Raimond, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 541–544.     

Équation des ondes sur les espaces symétriques riemanniens

We prove that the solutions of the homogeneous wave equation on Riemannian symmetric spaces have dispersion properties and we deduce Strichartz type estimates for these solutions. To cite this article: A. Hassani, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Transformation aux rayons X sur un espace symétrique

An inversion formula is proved for the X-ray transform on a Riemannian symmetric space of the non-compact type, by means of the shifted dual Radon transform. To cite this article: F. Rouvière, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Comparaison des volumes des variétés riemanniennes

Using the rigidity result of Besson, Courtois and Gallot, and also the notion of intersection of metrics, we compare volumes of Riemannian manifolds by means of lengths of their periodic geodesics. To cite this article: H.-R. Fanaï, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Compactness of solutions to the Yamabe problem

We establish compactness of solutions to the Yamabe problem on any smooth compact connected Riemannian manifold (not conformally diffeomorphic to standard spheres) of dimension n?7 as well as on any manifold of dimension n?8 under some additional hypothesis. To cite this article: Y.Y. Li, L. Zhang, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A sharp Sobolev inequality on Riemannian manifoldsUne inégalité de Sobolev optimale sur les varietés riemanniennes

We outline our results in [11] concerning some sharp Sobolev inequalities on Riemannian manifolds. Our inequalities emphasize the role of scalar curvature in this context. To cite this article: Y.Y. Li, T. Ricciardi, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 519–524.     

Backward volume contraction for endomorphisms with eventual volume expansion

We consider smooth maps on compact Riemannian manifolds. We prove that under some mild condition of eventual volume expansion Lebesgue almost everywhere we have uniform backward volume contraction on every pre-orbit of Lebesgue almost every point. To cite this article: J.F. Alves et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

A stochastic differential equation from friction mechanics

The existence and uniqueness of solutions to multivalued stochastic differential equations of the second order on Riemannian manifolds are proved. The class of problem is motivated by rigid body and multibody dynamics with friction and an application to the spherical pendulum with friction is presented. To cite this article: F. Bernardin et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On some conformally invariant fully nonlinear equationsSur certaines équations completement nonlinéaires invariantes par transformation conforme

We outline proofs of our results in [7] on Liouville type theorems, Harnack type inequalities, and existence and compactness of solutions to some conformally invariant fully nonlinear elliptic equations of second order on locally conformally flat Riemannian manifolds. Details will appear in [7]. To cite this article: A. Li, Y.Y. Li, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 305–310.