{"paper":{"title":"Sobolev Metrics on Diffeomorphism Groups and the Derived Geometry of Spaces of Submanifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"David Mumford, Mario Micheli, Peter W. Michor","submitted_at":"2012-02-16T19:36:50Z","abstract_excerpt":"Given a finite dimensional manifold $N$, the group $\\operatorname{Diff}_{\\mathcal S}(N)$ of diffeomorphism of $N$ which fall suitably rapidly to the identity, acts on the manifold $B(M,N)$ of submanifolds on $N$ of diffeomorphism type $M$ where $M$ is a compact manifold with $\\dim M<\\dim N$. For a right invariant weak Riemannian metric on $\\operatorname{Diff}_{\\mathcal S}(N)$ induced by a quite general operator $L:\\frak X_{\\mathcal S}(N)\\to \\Gamma(T^*N\\otimes\\operatorname{vol}(N))$, we consider the induced weak Riemannian metric on $B(M,N)$ and we compute its geodesics and sectional curvature."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.3677","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"}