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 ≤ 2qd ≤ 2nd, where q is the smallest value of i such that the ith homotopy group of the manifold is non-trivial.

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