Commentarii Mathematici Helvetici
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Published online: 2013-04-24
Vertices of closed curves in Riemannian surfacesMohammad Ghomi (1) Georgia Institute of Technology, Atlanta, United States
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.
Keywords: Four vertex theorem, Riemannian surface, surface of constant curvature, space form, hyperbolic surface, geodesic curvature
Ghomi Mohammad: Vertices of closed curves in Riemannian surfaces. Comment. Math. Helv. 88 (2013), 427-448. doi: 10.4171/CMH/290