Rigidity of compact manifolds with negative curvature (Q2366268)

This article is a survey of results that relate the smoothness of Anosov flows on a compact manifold to the rigidity of various geometric and dynamical properties of the manifold. One of the most important examples of an Anosov flow is the geodesic flow in the unit tangent bundle \(SM\) of a compact Riemannian manifold \(M\) with strictly negative sectional curvature. Geodesic flows are also examples of contact flows, flows that leave invariant a 1-form \(\theta\) on a manifold of dimension \(2n+1\) such that \(\theta\wedge(d\theta)^ n\) is a nowhere vanishing \((2n+1)\)-form. In general the stable and unstable Anosov foliations are absolutely continuous but not \(C^ 1\) although individual leaves of these foliations are \(C^ \infty\). If \(M\) is a compact manifold with negative sectional curvature that is pointwise strictly \(\alpha^ 2/4\) pinched for some \(\alpha\in(0,2]\), then the Anosov foliations of the geodesic flow in \(SM\) are \(C^{\alpha-\varepsilon}\) for every \(\varepsilon>0\) by a result of Hasselblatt. This generalizes an earlier result of Hirsch- Pugh-Shub and L. Green that the Anosov foliations are \(C^ 1\) if the sectional curvature is strictly quarter pinched. If an Anosov flow is sufficiently differentiable, then one expects some kind of rigidity and the article presents several examples. A result of Hurder and Katok shows that if one considers the geodesic flow in \(SM\) of a compact negatively curved surface \(M\), then the Anosov foliations are \(C^{1,1}\) if and only if \(M\) has constant negative curvature (in this case the foliations are \(C^ \infty)\). If \(M\) is a compact manifold of arbitrary dimension with sectional curvature \(K<0\) whose Anosov foliations in \(SM\) are \(C^{2,\alpha}\) for some \(\alpha>0\), then a result of Hamenstädt states that the Lebesgue, Bowen-Margulis and harmonic measure classes are equal. These measure classes are equal if \(M\) is locally symmetric, and conversely it may be the case that \(M\) is locally symmetric if any two of these measure classes are equal. If \(\{\varphi_ t\}\) is a contact Anosov flow on a compact manifold \(N\) whose Anosov foliations are \(C^ \infty\), then Y. Benoist, P. Foulon and F. Labourie have shown that after lifting the flow to a finite cover \(N^*\) of \(N\) the resulting flow \(\{\varphi^*_ t\}\) is \(C^ \infty\) time preserving conjugate to the geodesic flow of a compact locally symmetric manifold \(M\) with \(K<0\). In dimension 3 this result was proved earlier by E. Ghys. In arbitrary dimensions the first result of this type was proved for geodesic flows by Kanai, and the (Kanai) connection that he introduced in the unit tangent bundle has continued to play a key role in all subsequent improvements. If \(M\) is a compact manifold with \(K<0\) whose Anosov foliations in \(SM\) are \(C^ \infty\), then one expects that \(M\) itself is locally symmetric. However, this result remains unproved after strong efforts by many people. If \(M\) is compact and locally symmetric with \(K<0\), then the Anosov foliations in \(SM\) for the geodesic flow are \(C^ \infty\), the topological entropy equals the metric entropy, horospheres have constant mean curvature and the measure classes \(\lambda,\mu,\nu\) of Lebesgue, Bowen-Margulis and harmonic measure coincide. Conversely, any one of these properties may characterize locally symmetric spaces in the class of compact manifolds with sectional curvature \(K<0\). Much research has been devoted to these questions, and affirmative answers exist in special cases. If \(M\) is 2-dimensional with \(K<0\), then Katok and Ledrappier have shown that \(M\) has constant curvature if any two of the measure classes above coincide. Hamenstädt has shown that if \(\lambda=\mu\), then \(\lambda=\mu=\nu\) for compact manifolds with \(K<0\). Kaimanovich has introduced a probabilistic (Kaimanovich) entropy for compact manifolds with \(K<0\), and Kaimanovich and Ledrappier have derived sharp inequalities relating this entropy to topological entropy and the first eigenvalue of the Laplacian. Equality in one of these inequalities implies that \(\mu=\nu\) or that horospheres have constant mean curvature.