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Commentarii Mathematici Helvetici


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Volume 88, Issue 2, 2013, pp. 427–448
DOI: 10.4171/CMH/290

Published online: 2013-04-24

Vertices of closed curves in Riemannian surfaces

Mohammad Ghomi[1]

(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