{"paper":{"title":"Causal Holography in Application to the Inverse Scattering Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.GT","authors_text":"Gabriel Katz","submitted_at":"2017-03-26T21:16:42Z","abstract_excerpt":"For a given smooth compact manifold $M$, we introduce an open class $\\mathcal G(M)$ of Riemannian metrics, which we call \\emph{metrics of the gradient type}. For such metrics $g$, the geodesic flow $v^g$ on the spherical tangent bundle $SM \\to M$ admits a Lyapunov function (so the $v^g$-flow is traversing). It turns out, that metrics of the gradient type are exactly the non-trapping metrics.\n  For every $g \\in \\mathcal G(M)$, the geodesic scattering along the boundary $\\partial M$ can be expressed in terms of the \\emph{scattering map} $C_{v^g}: \\partial_1^+(SM) \\to \\partial_1^-(SM)$. It acts f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.08874","kind":"arxiv","version":4},"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"}