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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Geodesic flow in certain manifolds without conjugate points


Author: Patrick Eberlein
Journal: Trans. Amer. Math. Soc. 167 (1972), 151-170
MSC: Primary 58E10; Secondary 53C20
MathSciNet review: 0295387
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Abstract: A complete simply connected Riemannian manifold H without conjugate points satisfies the uniform Visibility axiom if the angle subtended at a point p by any geodesic $ \gamma $ of H tends uniformly to zero as the distance from p to $ \gamma $ tends uniformly to infinity. A complete manifold M is a uniform Visibility manifold if it has no conjugate points and if the simply connected covering H satisfies the uniform Visibility axiom. We derive criteria for the existence of uniform Visibility manifolds. Let M be a uniform Visibility manifold, SM the unit tangent bundle of M and $ {T_t}$ the geodesic flow on SM. We prove that if every point of SM is nonwandering with respect to $ {T_t}$ then $ {T_t}$ is topologically transitive on SM. We also prove that if $ M'$ is a normal covering of M then $ {T_t}$ is topologically transitive on $ SM'$ if $ {T_t}$ is topologically transitive on SM.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0295387-4
PII: S 0002-9947(1972)0295387-4
Keywords: Geodesic flow, conjugate points, nonwandering points, topological transitivity, uniform Visibility
Article copyright: © Copyright 1972 American Mathematical Society