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

• 成果类型: Article
• 署名作者: Radeschi, Marco; Wilking, Burkhard
• 署名单位: University of Munster
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-017-0742-4
• 发表日期: 2017
• 页码: 911-962
• 关键词: numbers; SPACES

摘要

We define a Besse manifold as a Riemannian manifold (M, g) 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.