American Institute of Mathematical Sciences

September  2020, 40(9): 5149-5171. doi: 10.3934/dcds.2020223

Contributions to the study of Anosov geodesic flows in non-compact manifolds

 1 Departamento de Matemática, Universidade Federal do Piauí, Centro de Ciências da Natureza - Avenida Universitária, Inanga, Teresina 64049-550, Brazil 2 Instituto de Matemática, Universidade Federal do Rio de Janeiro, Cidade Universitária - Ilha do Fundão, Rio de Janeiro 21941-909, Brazil

* Corresponding author: Sergio Romaña

Received  May 2019 Revised  January 2020 Published  June 2020

Fund Project: Ítalo Melo were partially supported by FAPEPI

In this paper, we study the relations between curvature and Anosov geodesic flow. More specifically, we prove that when the geodesic flow of a complete manifold without conjugate points is of the Anosov type, then the average of the sectional curvature in tangent planes along geodesics is negative and uniformly away from zero. Moreover, if a surface has no focal points, then the latter condition is sufficient to obtain that the geodesic flow is of Anosov type.

Citation: Ítalo Melo, Sergio Romaña. Contributions to the study of Anosov geodesic flows in non-compact manifolds. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5149-5171. doi: 10.3934/dcds.2020223
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