According to the classical result of J. P. Serre ([S]) any two points on a closed Riemannian manifold can be connected by infinitely many geodesics. The length of a shortest of them trivially does not exceed the diameter d of the manifold. But how long are the shortest remaining geodesics? In this paper we prove that any two points on a closed n-dimensional Riemannian manifold can be connected by two distinct geodesics of length ≤ 2qd ≤ 2nd, where q is the smallest value of i such that the ith homotopy group of the manifold is non-trivial.
Cite this article
Alexander Nabutovsky, Regina Rotman, The length of the second shortest geodesic. Comment. Math. Helv. 84 (2009), no. 4, pp. 747–755DOI 10.4171/CMH/179