Geometric conditions to obtain Anosov geodesic flow for non-compact manifolds

Abstract

Let $(M,g)$ be a complete Riemannian manifold without focal points and curvature bounded below. We prove that when the average of the sectional curvature in tangent planes along geodesics is negative and uniformly away from zero, then the geodesic flow is of Anosov type. We use this result to construct a non-compact manifold with non-positive curvature, without compact quotient and geodesic flow of Anosov type.

Keywords:

• Anosov geodesic flow
• Jacobi field
• focal points
• warped product

Mathematics Subject Classification (2010):

• 37D40
• 53C20

Acknowledgments

Alexander Cantoral thanks FAPERJ for partially supporting this research and The Institute of Mathematics of the Federal University of Rio de Janeiro (IM-UFRJ). Sergio Romaña thanks Department of Mathematics of the SUSTech- China for its hospitality during the execution of this work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was partially supported by the Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ - Brazil) [scholarship E-26/202.303/2022]. The second author was partially supported by program CAPES-PrInt-BR [process number 88887.311615/2018-00], Bolsa Jovem Cientista do Nosso Estado [grant number E-26/201.432/2022] – Brazil, National Natural Science Foundation of China (NNSFC) [grant number 12071202], and National Natural Science Foundation of China (NNSFC) [grant number 12161141002], Conselho Nacional de Desenvolvimento Científico e Tecnológico.