JournalscmhVol. 88, No. 2pp. 427–448

Vertices of closed curves in Riemannian surfaces

  • Mohammad Ghomi

    Georgia Institute of Technology, Atlanta, United States
Vertices of closed curves in Riemannian surfaces cover
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Abstract

We uncover some connections between the topology of a complete Riemannian surface MM and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in MM. 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. 427–448

DOI 10.4171/CMH/290