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 $\le 2qd\le 2nd$, where $q$ is the smallest value of $i$ such that the $i$th homotopy group of the manifold is non-trivial.