Linear bounds for lengths of geodesic loops on Riemannian 2-spheres. (English) Zbl 1243.53075

Nabutovsky, Alexander; Rotman, Regina

J. Differ. Geom. 89, No. 2, 217-232 (2011).

Summary: Let \(M\) be a closed surface diffeomorphic to \(S^2\) endowed with a Riemannian metric. Denote the diameter of \(M\) by \(d\). We prove that for every \(x \in M\) and every positive integer \(k\) there exist \(k\) distinct geodesic loops based at \(x\) of length \(\leq 20kd\).

MSC:

53C22

Geodesics in global differential geometry

Keywords:

diameter; geodesic loop; length