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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.

MSC:

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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