CRITICAL VALUES OF HOMOLOGY CLASSES OF LOOPS AND POSITIVE CURVATURE

We study compact and simply-connected Riemannian manifolds (M, g) with positive sectional curvature K >= 1. For a nontrivial homology class of lowest positive dimension in the space of loops based at a point p is an element of M or in the free loop space one can define a critical length crl(p) (M, g) resp. crl (M, g). Then crl(p) (M, g) equals the length of a geodesic loop with base point p and crl (M, g) equals the length of a closed geodesic. This is the idea of the proof of the existence of a closed geodesic of positive length presented by Birkhoff in case of a sphere and by Lusternik & Fet in the general case. It is the main result of the paper that the numbers crl(p) (M, g) resp. crl (M, g) attain its maximal value 2 pi only for the round metric on the n-sphere. Under the additional assumption K <= 4 this result for crl (M, g) follows from results by Sugimoto in even dimensions and Ballmann, Thorbergsson & Ziller in odd dimensions.