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On problems of systolic geometry. (Sur des problèmes de la géométrie systolique.) (French) Zbl 1083.53043
Seminar on spectral theory and geometry. 2003–2004. St. Martin d’Hères: Université de Grenoble I, Institut Fourier. Séminaire de Théorie Spectrale et Géométrie 22, 71-82 (2004).
The author presents an universal upper bound for the length of a shortest nontrivial geodesic loop in terms of the diameter for a simply-connected closed Riemannian manifold with non-trivial second homotopy group. Also, a short overview on previous results by M. Maeda, C. Croke, A. Nabutovsky/R. Rotman and S. Sabourau is given. In the second section the systolic constant of a closed surface is estimated from below by a function of the genus. This is related to results by M. Gromov and P. Buser/P. Sarnak.
For the entire collection see [Zbl 1054.35001].

MSC:
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C22 Geodesics in global differential geometry
05C10 Planar graphs; geometric and topological aspects of graph theory
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