# Commentarii Mathematici Helvetici

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

**DOI: 10.4171/CMH/179**

The length of the second shortest geodesic

Alexander Nabutovsky^{[1]}and Regina Rotman

^{[2]}(1) Department of Mathematics, University of Toronto, 100 St. George Street, M5S 3G3, TORONTO, ONTARIO, CANADA

(2) Department of Mathematics, University of Toronto, 100 St. George Street, M5S 3G3, TORONTO, ONTARIO, 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