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**Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations.**
*(English)*
Zbl 0634.58020

Suppose that M is a closed Riemannian manifold of negative curvature. The geodesic flow \(\phi\) \(t_ M\) of M, which is defined as a smooth flow on the unit tangent bundle \(V_ M=\{v\in TM:| v| =1\}\) of M, is known to be Anosov. In other words, the tangent bundle \(TV_ M\) of the phase space \(V_ M\) admits a unique \(\phi\) \(t_ M\)-invariant continuous splitting \(TV_ M=E\) \(-+E\) \(0+E\) \(+\) into subbundles satisfying the following conditions: (i) E 0 is of dimension one and tangent to the flow; (ii) For each \(\xi ^{\pm}\in E^{\pm}\), \(d\phi\) \(t_ M\xi ^{\pm}\) contracts exponentially as \(t\to \pm \infty\). We call this splitting the Anosov splitting of M, and the purpose of the paper is to indicate an obstruction to the smoothness of the Anosov splitting.

More precisely we prove the following theorem. Let M be a closed Riemannian manifold of dimension \(\geq 3\) whose sectional curvature K satisfies the pinching condition \(-9/4<K\leq -1\). If the Anosov splitting of M is \(C^{\infty}\), then the geodesic flow of M is isomorphic to that of a certain closed Riemannian manifold of constant negative curvature in the sense that there exists a \(C^{\infty}\) diffeomorphism \(f: V_ M\to V_ N\) such that \(f\circ \phi\) \(t_ M=\phi\) \(t_ N\circ f\) for all \(t\in {\mathbb{R}}\).

More precisely we prove the following theorem. Let M be a closed Riemannian manifold of dimension \(\geq 3\) whose sectional curvature K satisfies the pinching condition \(-9/4<K\leq -1\). If the Anosov splitting of M is \(C^{\infty}\), then the geodesic flow of M is isomorphic to that of a certain closed Riemannian manifold of constant negative curvature in the sense that there exists a \(C^{\infty}\) diffeomorphism \(f: V_ M\to V_ N\) such that \(f\circ \phi\) \(t_ M=\phi\) \(t_ N\circ f\) for all \(t\in {\mathbb{R}}\).

Reviewer: M.Kanai

### MSC:

37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |

53D25 | Geodesic flows in symplectic geometry and contact geometry |

37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |

53C22 | Geodesics in global differential geometry |

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\textit{M. Kanai}, Ergodic Theory Dyn. Syst. 8, No. 2, 215--239 (1988; Zbl 0634.58020)

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