{"paper":{"title":"Condition metrics in the three classical spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Juan G. Criado del Rey","submitted_at":"2015-01-19T11:25:12Z","abstract_excerpt":"Let $(\\mathcal{M},g)$ be a Riemannian manifold and $\\mathcal{N}$ a $\\mathcal{C}^2$ submanifold without boundary. If we multiply the metric $g$ by the inverse of the squared distance to $\\mathcal{N}$, we obtain a new metric structure on $\\mathcal{M}\\setminus\\mathcal{N}$ called the condition metric. A question about the behaviour of the geodesics in this new metric arises from the works of Shub and Beltr\\'an: is it true that for every geodesic segment in the condition metric its closest point to $\\mathcal{N}$ is one of its endpoints? Previous works show that the answer to this question is positi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.04456","kind":"arxiv","version":1},"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"}