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Inventiones mathematicae

, Volume 210, Issue 3, pp 911–962 | Cite as

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

  • Marco Radeschi
  • Burkhard Wilking
Article

Abstract

We define a Besse manifold as a Riemannian manifold (Mg) all of whose geodesics are closed. A conjecture of Berger states that all prime geodesics have the same length for any simply connected Besse manifold. We firstly show that the energy function on the free loop space of a simply connected Besse 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.

Mathematics Subject Classification

53C22 58E10 

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.University of MünsterMünsterGermany

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