Local lens rigidity for manifolds of Anosov type

2022/04/05 by Cekić, Mihajlo, Guillarmou, Colin, Lefeuvre, Thibault
#35R30 #53C24 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2204.02476

Abstract

The lens data of a Riemannian manifold with boundary is the collection of lengths of geodesics with endpoints on the boundary together with their incoming and outgoing vectors. We show that negatively-curved Riemannian manifolds with strictly convex boundary are locally lens rigid in the following sense: if g₀ is such a metric, then any metric g sufficiently close to g₀ and with same lens data is isometric to g₀, up to a boundary-preserving diffeomorphism. More generally, we consider the same problem for a wider class of metrics with strictly convex boundary, called metrics of Anosov type. We prove that the same rigidity result holds within that class in dimension 2 and in any dimension, further assuming that the curvature is non-positive.

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