## Geodesic flows on negatively curved manifolds. II

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- by Patrick Eberlein PDF
- Trans. Amer. Math. Soc.
**178**(1973), 57-82 Request permission

## Abstract:

Let*M*be a complete Riemannian manifold with sectional curvature $K \leq 0$,

*SM*the unit tangent bundle of

*M*, ${T_t}$ the geodesic flow on

*SM*and $\Omega \subseteq SM$ the set of nonwandering points relative to ${T_t}$. ${T_t}$ is

*topologically mixing*(respectively

*topologically transitive*) on

*SM*if for any open sets

**0**,

*U*of

*SM*there exists $A > 0$ such that $\left | t \right | \geq A$ implies ${T_t}(O) \cap U \ne \emptyset$ (respectively there exists $t\;\varepsilon \;R$ such that ${T_t}(O) \cap U \ne \emptyset$). For each vector $v\;\varepsilon \;SM$ we define stable and unstable sets ${W^s}(v),{W^{ss}}(v),{W^u}(v)$ and ${W^{uu}}(v)$, and we relate topological mixing (respectively topological transitivity) of ${T_t}$ to the existence of a vector $v\; \in \;SM$ such that ${W^{ss}}(v)$ (respectively ${W^s}(v)$) is dense in

*SM*. If

*M is a Visibility manifold*(implied by $K \leq c < 0$) and if $\Omega = SM$ then ${T_t}$ is topologically mixing on

*SM*. Let ${S_n} =$ {Visibility manifolds

*M*of dimension

*n*such that ${T_t}$ is topologically mixing on

*SM*}. For each $n \geq 2$, ${S_n}$ is closed under normal (Galois) Riemannian coverings. If $M\; \in \;{S_n}$ we classify {$v\; \in \;SM:\;{W^{ss}}(v)$ is dense in

*SM*}, and

*M*is compact if and only if this set =

*SM*. We also consider the case where $\Omega$ is a proper subset of

*SM*.

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## Additional Information

- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**178**(1973), 57-82 - MSC: Primary 58F10; Secondary 53C20, 53C70
- DOI: https://doi.org/10.1090/S0002-9947-1973-0314084-0
- MathSciNet review: 0314084