Le flot géodésique des variétés riemanniennes à courbure négative
Le spectre des longueurs des surfaces hyperboliques : un exemple de rigidité.
Après avoir présenté quelques résultats récents portant sur l’étude du spectre des longueurs des surfaces hyperboliques avec ou sans singularités, on démontre que les sphères possédant trois points coniques sont, dans leur classe, spectralement rigides.
Le spectre marqué des longueurs des surfaces à courbure négative
Le théorème de Gromoll-Meyer sur les géodésiques fermées
Length of curves on Lip manifolds
In this paper the length of a curve on a Lipschitz Riemannian manifold is defined. It is shown that the above definition is consistent with the definition of the geodesic distance already introduced by the authors, both in a geometrical and analytical way.
Length spectrum invariants of Riemannian manifolds.
Les groupes de triangles sont déterminés par leur spectre des longueurs
On décrit le début du spectre des longueurs des groupes de triangles ayant un angle droit et on montre que le spectre des longueurs caractérise la classe d’isométrie d’un tel groupe.
Long geodesics on convex surfaces.
Lorentzian geometry in the large
Lorentzian geometry in the large has certain similarities and certain fundamental differences from Riemannian geometry in the large. The Morse index theory for timelike geodesics is quite similar to the corresponding theory for Riemannian manifolds. However, results on completeness for Lorentzian manifolds are quite different from the corresponding results for positive definite manifolds. A generalization of global hyperbolicity known as pseudoconvexity is described. It has important implications...
Magnetic Curves in a Euclidean Space: One Example, Several Approaches
Manifolds with Geodesic Chords of Constant Length.
Measured geodesic laminations in Flatland
Since their introduction by Thurston, measured geodesic laminations on hyperbolic surfaces occur in many contexts. In this survey, we give a generalization of geodesic laminations on surfaces endowed with a half-translation structure (that is a singular flat surface with holonomy ), called flat laminations, and we define transverse measures on flat laminations similar to transverse measures on hyperbolic laminations, taking into account that the images of the leaves of a flat lamination are in...
Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-riemannienne dans le cas Martinet
Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-riemannienne dans le cas Martinet
Consider a sub-riemannian geometry(U,D,g) where U is a neighborhood of 0 in R3, D is a Martinet type distribution identified to ker ω, ω being the 1-form: , q=(x,y,z) and g is a metric on D which can be taken in the normal form: , a=1+yF(q), c=1+G(q), . In a previous article we analyze the flat case: a=c=1; we describe the conjugate and cut loci, the sphere and the wave front. The objectif of this article is to provide a geometric and computational framework to analyze the general case....
Metric Entropy of Homogeneous Spaces
For a precompact subset K of a metric space and ε > 0, the covering number N(K,ε) is defined as the smallest number of balls of radius ε whose union covers K. Knowledge of the metric entropy, i.e., the asymptotic behaviour of covering numbers for (families of) metric spaces is important in many areas of mathematics (geometry, functional analysis, probability, coding theory, to name a few). In this paper we give asymptotically correct estimates for covering numbers for a large class of homogeneous...
Metric with ergodic geodesic flow is completely determined by unparameterized geodesics.
Metrics in the sphere of a C*-module
Given a unital C*-algebra and a right C*-module over , we consider the problem of finding short smooth curves in the sphere = x ∈ : 〈x, x〉 = 1. Curves in are measured considering the Finsler metric which consists of the norm of at each tangent space of . The initial value problem is solved, for the case when is a von Neumann algebra and is selfdual: for any element x 0 ∈ and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ , Z* = −Z and ∥Z∥ ≤ π, such...
Metrics with homogeneous geodesics on flag manifolds
A geodesic of a homogeneous Riemannian manifold is called homogeneous if it is an orbit of an one-parameter subgroup of . In the case when is a naturally reductive space, that is the -invariant metric is defined by some non degenerate biinvariant symmetric bilinear form , all geodesics of are homogeneous. We consider the case when is a flag manifold, i.eȧn adjoint orbit of a compact semisimple Lie group , and we give a simple necessary condition that admits a non-naturally reductive...
Metrics with only finitely many isometry invariant geodesics.
Monge-Ampère Equations, Geodesics and Geometric Invariant Theory
Existence and uniqueness theorems for weak solutions of a complex Monge-Ampère equation are established, extending the Bedford-Taylor pluripotential theory. As a consequence, using the Tian-Yau-Zelditch theorem, it is shown that geodesics in the space of Kähler potentials can be approximated by geodesics in the spaces of Bergman metrics. Motivation from Donaldson’s program on constant scalar curvature metrics and Yau’s strategy of approximating Kähler metrics by Bergman metrics is also discussed....