A min-max characterization of Zoll Riemannian metrics

Abstract

We characterize the Zoll Riemannian metrics on a given simply connected spin closed manifold as those Riemannian metrics for which two suitable min-max values in a finite dimensional loop space coincide. We also show that on odd dimensional Riemannian spheres, when certain pairs of min-max values in the loop space coincide, every point lies on a closed geodesic.

24 pages, 1 figure; version 2: Theorem 1.1 now stated for manifolds homeomorphic to odd-dimensional spheres (the original statement was correct, but indeed the assumptions made on the manifold forced it to be a topological sphere according to Bott-Samelson theorem and the Poincaré conjecture); minor correction in the centered equation before (5.5)

References in corpus (2)

• A characterization of Zoll Riemannian metrics on the 2-sphere
• On the spectral characterization of Besse and Zoll Reeb flows

Cited by in corpus (2)

• The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed
• On the structure of Besse convex contact spheres