Curve shortening and the topology of closed geodesics on surfaces. (English) Zbl 1137.53330

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.


53C22 Geodesics in global differential geometry
58D10 Spaces of embeddings and immersions
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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