2014/12/04 by Colin Guillarmou, Guillarmou, Colin · 2 citations
Mathematics · #35R30 #37D20 #37D40 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Morphological variations and asymmetry #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1412.1760
openalex publication_date 2014/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a Riemannian manifold (M,g) with strictly convex boundary ∂ M, the lens data consists in the set of lengths of geodesics γ with endpoints on ∂ M, together with their endpoints (x-,x+)∈ ∂ M× ∂ M and tangent exit vectors (v-,v+)∈ Tx- M× Tx+ 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.