Prime geodesic theorem.
Proximinality in geodesic spaces.
Quadratic Divergence of Geodesics in CAT (0) Spaces.
Quand deux sous-variétés sont forcées par leur géométrie à se rencontrer
Quasiconformality in the Geodesic Flow of Negatively Curved Manifolds.
Rank and symmetry of Riemannian manifolds.
Real hypersurfaces in a nonflat complex space form and their almost contact metric structures
We characterize homogeneous real hypersurfaces of types (A₀), (A₁) and (B) in a complex projective space or a complex hyperbolic space.
Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I : regularity
We formulate an Hamilton-Jacobi partial differential equationon a dimensional manifold , with assumptions of convexity of and regularity of (locally in a neighborhood of in ); we define the “min solution” , a generalized solution; to this end, we view as a symplectic manifold. The definition of “min solution” is suited to proving regularity results about ; in particular, we prove in the first part that the closure of the set where is not regular may be covered by a countable number...
Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I: Regularity (errata)
This errata corrects one error in the 2004 version of this paper [Mennucci, ESAIM: COCV10 (2004) 426–451].
Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I: Regularity
We formulate an Hamilton-Jacobi partial differential equation H( x, D u(x))=0 on a n dimensional manifold M, with assumptions of convexity of H(x, .) and regularity of H (locally in a neighborhood of {H=0} in T*M); we define the “minsol solution” u, a generalized solution; to this end, we view T*M as a symplectic manifold. The definition of “minsol solution” is suited to proving regularity results about u; in particular, we prove in the first part that the closure of the set where...
Regularity properties of the distance functions to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and applications in Riemannian geometry
Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, we show that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka [Trans. Amer. Math. Soc. 353 (2001) 21–40],...
Relaxed hyperelastic curves
We define relaxed hyperelastic curve, which is a generalization of relaxed elastic lines, on an oriented surface in three-dimensional Euclidean space E³, and we derive the intrinsic equations for a relaxed hyperelastic curve on a surface. Then, by examining relaxed hyperelastic curves in a plane, on a sphere and on a cylinder, we show that geodesics are relaxed hyperelastic curves in a plane and on a sphere. But on a cylinder, they are relaxed hyperelastic curves only in special cases.
Remarks on F-planar curves and their generalizations
Generalized planar curves (A-curves) are more general analogues of F-planar curves and geodesics. In particular, several well known geometries are described by more than one affinor. The best known example is the almost quaternionic geometry. A new approach to this topic (A-structures) was started in our earlier papers. In this paper we expand the concept of A-structures to projective A-structures.
Representation of Markoff's binary quadratic forms by geodesics on a perforated torus
Restriction de la distance géodésique à un arc et rigidité
Riemann Surfaces with Shortest Geodesic of Maximal Length.
Riemannsche Flächen mit grosser Kragenweite.
Rigidité du flot géodésique de certaines nilvariétés de rang deux
Rigidity and sphere theorem for manifolds with positive Ricci Curvature.
Rigidity for surfaces of non-positive curvature.