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