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

DOI:
https://doi.org/10.1090/S0002-9947-1972-0295387-4

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 of *H* tends uniformly to zero as the distance from *p* to 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 the geodesic flow on *SM*. We prove that if every point of *SM* is nonwandering with respect to then is topologically transitive on *SM*. We also prove that if is a normal covering of *M* then is topologically transitive on if is topologically transitive on *SM*.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0295387-4

Keywords:
Geodesic flow,
conjugate points,
nonwandering points,
topological transitivity,
uniform Visibility

Article copyright:
© Copyright 1972
American Mathematical Society