Vertices of closed curves in Riemannian surfaces

Details Vertices of closed curves in Riemannian surfaces Vertices of closed curves in Riemannian surfaces

2010-06-21

arxiv.org/abs/1006.4182v1

Mathematics

Differential Geometry

19 pages, 10 figures

Scientific paper

We uncover some connections between the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in M. In particular we show that the space forms with finite fundamental group are the only surfaces in which every simple closed curve has more than two vertices. Further we characterize the simply connected space forms as the only surfaces in which every closed curve bounding a compact immersed surface has more than two vertices.

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