{"paper":{"title":"Laplace Transformations of Submanifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Bang-Yen Chen, Leopold Verstraelen","submitted_at":"2013-07-05T03:46:51Z","abstract_excerpt":"Let $x : M \\to E^m$ be an isometric immersion of a Riemannian manifold $M$ into a Euclidean $m$-space. Denote by $\\Delta$ the Laplace operator of $M$. Then $\\Delta$ gives rise to a differentiable map $L :M \\to E^m$, called the Laplace map, defined by $L(p)=(\\Delta x)(p)$, $p\\in M$. We call $L(M)$ the Laplace image, and the transformation $L :M \\to L(M)$ from $M$ onto its Laplace image $L(M)$ the {\\it Laplace transformation}. In this monograph, we provide a fundamental study of the Laplace transformations of Euclidean submanifolds."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.1515","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"}