On the Berger conjecture for manifolds all of whose geodesics are closed

2015/11/24 by Marco Radeschi, Radeschi, Marco, Burkhard Wilking +1
Mathematics · #53C22 #58E10 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C22 #msc:58E10

paper · pdf · doi:10.48550/arxiv.1511.07852

39 pages

arxiv created 2015/11/24 · arxiv updated 2015/11/25

Abstract

A conjecture of Berger states that, for any simply connected Riemannian manifold all of whose geodesics are closed, all prime geodesics have the same length. We firstly show that the energy function on the free loop space of such a manifold is a perfect Morse-Bott function with respect to a suitable cohomology. Secondly we explain when the negative bundles along the critical manifolds are orientable. These two general results then lead to a solution of Berger's conjecture when the underlying manifold is a sphere of dimension at least four.

Related

• Three dimensional manifolds all of whose geodesics are closed
• The Existence of Infinitely Many Geometrically Distinct Non-Constant Prime Closed Geodesics on Riemannian Manifolds
• Morse theory, closed geodesics, and the homology of free loop spaces
• Closed geodesics on orbifolds
• Critical values of homology classes of loops and positive curvature