The length of the second shortest geodesic

Abstract

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.