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

Ítalo Melo; Sergio Romaña

doi:10.3934/dcds.2020223

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2020, Volume 40, Issue 9: 5149-5171. Doi: 10.3934/dcds.2020223

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

Ítalo Melo^1, and
Sergio Romaña^2, ,

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
^* Corresponding author: Sergio Romaña

Received: May 2019

Revised: January 2020

Published: June 2020

Ítalo Melo were partially supported by FAPEPI

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Abstract

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.

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Mathematics Subject Classification: Primary: 37D40, 37D20; Secondary: 53C21.

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References

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