We uncover some connections between the topology of a complete Riemannian surface and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in . 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.
Cite this article
Mohammad Ghomi, Vertices of closed curves in Riemannian surfaces. Comment. Math. Helv. 88 (2013), no. 2, pp. 427–448DOI 10.4171/CMH/290