Angenent, Sigurd B. Curve shortening and the topology of closed geodesics on surfaces. (English) Zbl 1137.53330 Ann. Math. (2) 162, No. 3, 1187-1241 (2005). Summary: We study “flat knot types” of geodesics on compact surfaces \(M^2\). For every flat knot type and any Riemannian metric \(g\) we introduce a Conley index associated with the curve shortening flow on the space of immersed curves on \(M^2\). We conclude existence of closed geodesics with prescribed flat knot types, provided the associated Conley index is nontrivial. Cited in 1 ReviewCited in 8 Documents MSC: 53C22 Geodesics in global differential geometry 58D10 Spaces of embeddings and immersions 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) Keywords:flat knot types; compact surfaces; Conley index PDF BibTeX XML Cite \textit{S. B. Angenent}, Ann. Math. (2) 162, No. 3, 1187--1241 (2005; Zbl 1137.53330) Full Text: DOI arXiv OpenURL