Lens rigidity for manifolds with hyperbolic trapped set

TL;DR

This paper proves that for certain negatively curved manifolds with convex boundary, the lens data uniquely determines the manifold's geometry and topology, even with trapped geodesics present.

Contribution

It establishes deformation lens rigidity for a broad class of manifolds, including those with negative curvature, non-trivial topology, and trapped geodesics, and in dimension two, it shows scattering data determines topology and conformal class.

Findings

Deformation lens rigidity holds for manifolds with negative curvature and convex boundary.

Scattering data determines topology and conformal class in 2D.

Results include manifolds with trapped geodesics.

Abstract

For a Riemannian manifold $(M,g)$ with strictly convex boundary $\partial M$, the lens data consists in the set of lengths of geodesics $\gamma$ with endpoints on $\partial M$, together with their endpoints $(x_-,x_+)\in \partial M\times \partial M$ and tangent exit vectors $(v_-,v_+)\in T_{x_-} M\times T_{x_+} M$. We show deformation lens rigidity for a large class of manifolds which includes all manifolds with negative curvature and strictly convex boundary, possibly with non-trivial topology and trapped geodesics. For the same class of manifolds in dimension $2$, we prove that the set of endpoints and exit vectors of geodesics (ie. the scattering data) determines the topology and the conformal class of the surface.