  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.
