{"paper":{"title":"Determination of a Riemannian manifold from the distance difference functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Matti Lassas, Teemu Saksala","submitted_at":"2015-10-21T07:50:37Z","abstract_excerpt":"Let $(N,g)$ be a Riemannian manifold with the distance function $d(x,y)$ and an open subset $M\\subset N$. For $x\\in M$ we denote by $D_x$ the distance difference function $D_x:F\\times F\\to \\mathbb R$, given by $D_x(z_1,z_2)=d(x,z_1)-d(x,z_2)$, $z_1,z_2\\in F=N\\setminus M$. We consider the inverse problem of determining the topological and the differentiable structure of the manifold $M$ and the metric $g|_M$ on it when we are given the distance difference data, that is, the set $F$, the metric $g|_F$, and the collection $\\mathcal D(M)=\\{D_x;\\ x\\in M\\}$. Moreover, we consider the embedded image "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.06157","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"}