{"paper":{"title":"Scattering rigidity with trapped geodesics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Christopher B. Croke","submitted_at":"2011-03-28T23:52:45Z","abstract_excerpt":"We prove that the flat product metric on $D^n\\times S^1$ is scattering rigid where $D^n$ is the unit ball in $\\R^n$ and $n\\geq 2$. The scattering data (loosely speaking) of a Riemannian manifold with boundary is map $S:U^+\\partial M\\to U^-\\partial M$ from unit vectors $V$ at the boundary that point inward to unit vectors at the boundary that point outwards. The map (where defined) takes $V$ to $\\gamma'_V(T_0)$ where $\\gamma_V$ is the unit speed geodesic determined by $V$ and $T_0$ is the first positive value of $t$ (when it exists) such that $\\gamma_V(t)$ again lies in the boundary. We show th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.5511","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}