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

TL;DR

This paper proves Berger's conjecture for higher-dimensional spheres by analyzing the Morse-Bott properties of the energy function on the free loop space and the orientability of negative bundles, confirming all prime geodesics have the same length.

Contribution

It establishes the Morse-Bott perfection of the energy function and the orientability conditions, leading to a proof of Berger's conjecture on spheres of dimension at least four.

Findings

Energy function is a perfect Morse-Bott function on the free loop space.

Negative bundles along critical manifolds are orientable under certain conditions.

Berger's conjecture holds for spheres of dimension ≥ 4.

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.