# The length of the second shortest geodesic

### Alexander Nabutovsky

University of Toronto, Canada### Regina Rotman

University of Toronto, Canada

## 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 $i$th 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. 747–755

DOI 10.4171/CMH/179