{"paper":{"title":"Limitations on the smooth confinement of an unstretchable manifold","license":"","headline":"","cross_cats":["math.DG","math.MP"],"primary_cat":"math-ph","authors_text":"E. M. Kramer, R. P. Geroch, S. C. Venkataramani, T. A. Witten","submitted_at":"2000-07-03T17:01:22Z","abstract_excerpt":"We prove that an m-dimensional unit ball D^m in the Euclidean space {\\mathbb R}^m cannot be isometrically embedded into a higher-dimensional Euclidean ball B_r^d \\subset {\\mathbb R}^d of radius r < 1/2 unless one of two conditions is met -- (1)The embedding manifold has dimension d >= 2m. (2) The embedding is not smooth. The proof uses differential geometry to show that if d<2m and the embedding is smooth and isometric, we can construct a line from the center of D^m to the boundary that is geodesic in both D^m and in the embedding manifold {\\mathbb R}^d. Since such a line has length 1, the dia"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0007003","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"}