{"paper":{"title":"Pointwise Characterizations of Curvature and Second Fundamental Form on Riemannian Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Bo Wu, Feng-Yu Wang","submitted_at":"2016-05-09T07:26:47Z","abstract_excerpt":"Let $M$ be a complete Riemannian manifold possibly with a boundary $\\partial M$. For any $C^1$-vector field $Z$, by using gradient/functional inequalities of the (reflecting) diffusion process generated by $L:=\\Delta+Z$, pointwise characterizations are presented for the Bakry-Emery curvature of $L$ and the second fundamental form of $\\partial M$ if exists. These extend and strengthen the recent results derived by A. Naber for the uniform norm $\\|{\\mathbf{Ric}}_Z\\|_\\infty$ on manifolds without boundary. A key point of the present study is to apply the asymptotic formulas for these two tensors f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.02447","kind":"arxiv","version":5},"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"}