# Commentarii Mathematici Helvetici

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**Volume 84, Issue 4, 2009, pp. 747–755**

**DOI: 10.4171/CMH/179**

Published online: 2009-12-23

The length of the second shortest geodesic

Alexander Nabutovsky^{[1]}and Regina Rotman

^{[2]}(1) University of Toronto, Canada

(2) University of Toronto, Canada

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 ≤ 2`q``d` ≤ 2`n``d`, where `q` is the smallest value of
`i` such that the `i`th homotopy group of the manifold is non-trivial.

*Keywords: *Length of geodesics, length functional, curvature-free inequalities in Riemannian geometry

Nabutovsky Alexander, Rotman Regina: The length of the second shortest geodesic. *Comment. Math. Helv.* 84 (2009), 747-755. doi: 10.4171/CMH/179