# Sur quelques problèmes de la géométrie des systoles

Abstract : This thesis is devoted to the study of universal geometric inequalities on Riemannian manifolds. More specifically, we focus our attention on free curvature relations between the volume and the length of short closed geodesics.

First, we study the extremal metrics of the isosystolic problem on surfaces. We establish a criterion for the extremality of metrics on orientable surfaces and examine the case of genus two.

Then, we show that the length of the shortest nontrivial trajectory among the simple closed geodesics of index one and the figure eight geodesics of null index bounds from below the area and the diameter of Riemannian spheres. We also discuss the rigidity and the softness of the filling radius with respect to the lengths of short closed geodesics arising from Morse theory on the one-cycle space.

Finally, we bound from below the volume and the diameter of complete Riemannian manifolds using the length of the shortest nontrivial geodesic loop. Furthermore, we obtain a lower bound on the growth of the volume of balls of small'' radius, as well as a homotopic finiteness result.
Mots-clés :
Document type :
Theses
Domain :

https://tel.archives-ouvertes.fr/tel-00001175
Contributor : Stéphane Sabourau Connect in order to contact the contributor
Submitted on : Thursday, February 28, 2002 - 5:00:54 PM
Last modification on : Wednesday, September 15, 2021 - 12:14:02 PM
Long-term archiving on: : Tuesday, September 11, 2012 - 5:00:41 PM

### Identifiers

• HAL Id : tel-00001175, version 1

### Citation

Stéphane Sabourau. Sur quelques problèmes de la géométrie des systoles. Mathématiques [math]. Université Montpellier II - Sciences et Techniques du Languedoc, 2001. Français. ⟨tel-00001175⟩

Record views