The marked length spectrum of Anosov manifolds. (English) Zbl 1506.53054
Summary: In all dimensions, we prove that the marked length spectrum of a Riemannian manifold \((M,g)\) with Anosov geodesic flow and non-positive curvature locally determines the metric in the sense that two close enough metrics with the same marked length spectrum are isometric. In addition, we provide a new stability estimate quantifying how the marked length spectrum controls the distance between the isometry classes of metrics. In dimension \(2\) we obtain similar results for general metrics with Anosov geodesic flows. We also solve locally a rigidity conjecture of Croke relating volume and marked length spectrum for the same category of metrics. Finally, by a compactness argument, we show that the set of negatively curved metrics (up to isometry) with the same marked length spectrum and with curvature in a bounded set of \(C^\infty\) is finite.
MSC:
| 53C24 | Rigidity results |
| 53C22 | Geodesics in global differential geometry |
| 37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |
| 37C27 | Periodic orbits of vector fields and flows |
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