{"corpus_id":47691293,"paper_sha":"c1d27c549ce9a9442d5578b466f22adeed41e584","doi":"10.1090/S0002-9904-1964-11062-4","arxiv_id":null,"pmid":null,"pmcid":null,"mag_id":1981760956,"dblp_id":null,"acl_id":null,"title":"A generalized Morse theory","year":1964,"publication_date":null,"venue":"","journal":{"name":"Bulletin of the American Mathematical Society","pages":"165-172","volume":"70"},"journal_issn":null,"journal_title":null,"publication_types":[],"pubmed_pub_types":null,"s2_fields_of_study":["Mathematics"],"reference_count":7,"citation_count":294,"influential_citation_count":31,"is_open_access":true,"arxiv_categories":null,"arxiv_license":null,"arxiv_journal_ref":null,"mesh_headings":null,"chemicals":null,"comments_corrections":null,"source_flags":1,"s2_open_access_pdf_url":"https://www.ams.org/bull/1964-70-01/S0002-9904-1964-11062-4/S0002-9904-1964-11062-4.pdf","s2_open_access_landing_url":"https://www.semanticscholar.org/paper/c1d27c549ce9a9442d5578b466f22adeed41e584","s2_open_access_license":null,"s2_open_access_status":"GOLD","pmc_open_access_pdf_url":null,"pmc_open_access_landing_url":null,"pmc_open_access_license":null,"pmc_open_access_status":null,"unpaywall_open_access_pdf_url":null,"unpaywall_open_access_landing_url":null,"unpaywall_open_access_license":null,"unpaywall_open_access_status":null,"abstract":"Let M be a C 2 -Riemannian manifold without boundary modeled on a separable Hubert space (see Lang [3]). For pzM we denote by ( , ) p the inner product in the tangent space M p and we define a function || || on the tangent bundle T(M) by ||z>|| = (v, v) x J 2 for vzMp. Given p and q in the same component of M we define p(p, q)==lnfl\\\\<r'(t)\\\\dt t where the Inf is over all C 1 paths <r\\ [0, l]->ikf such that a(0)=p and cr(l)=g. Just as in the finite dimensional case one shows that p is a metric on each component of M which is consistent with the manifold topology. If each component of M is complete in this metric M is called a complete Riemannian manifold and we assume this in all that follows. Let : M->R be a C 2 function. Then df, the differential of/, is a C 1 cross section of the cotangent bundle of M, hence there is a uniquely determined C 1 vector field V/ on ikf, the gradient of/, such that df p (v) = (#, *7f(p))p for v(~M p . We denote by $* the maximum local oneparameter group generated by -V/. A critical point of is a point where V/ vanishes; equivalently a stationary point of <*. At a critical point p of there is a uniquely determined continuous bilinear form H(f) p on M P1 the Hessian of at p } such that H(f) p (u, v) = d 2 (/ o <jr l )(d<}>p(u), d<p p (v)) if < is any chart at p. The supremum of the dimensions of subspaces on which H(f) p is negative (positive) definite is called the index (coindex) of at p. H(f) p is symmetric, hence there is a uniquely determined bounded self-ad joint operator A on M p such that H{f) v {u 1 v) = (Au, v) p . The critical point p is called nondegenerate if A has a bounded inverse. In this case p is isolated in the set of critical points.","claims":[{"public_id":"cl_43aa4c58aa5a2c59745c6914fea7bc14","status":"active","text":"A complete Riemannian manifold without boundary modeled on a separable Hilbert space has a metric consistent with the manifold topology defined by the infimum of lengths of C¹ paths between points.","confidence":0.95,"contributors":[{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["extraction"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_43aa4c58aa5a2c59745c6914fea7bc14"},{"public_id":"cl_11016575b01f11c8c5254287edd74146","status":"active","text":"A critical point p is nondegenerate if the bounded self-adjoint operator A on M_p representing H(f)_p has a bounded inverse; such a point is isolated in the set of critical points.","confidence":0.9,"contributors":[{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["extraction"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_11016575b01f11c8c5254287edd74146"},{"public_id":"cl_eb426ef1ae6821f0a627dfcd94d0b26a","status":"active","text":"At a critical point p of f, the Hessian H(f)_p is a continuous symmetric bilinear form on the tangent space M_p, and the index (coindex) is the supremum of dimensions of subspaces on which H(f)_p is negative (positive) definite.","confidence":0.95,"contributors":[{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["extraction"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous 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