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Linear bounds for the lengths of geodesics on manifolds with curvature bounded below
Beach, Isabel Contreras-Peruyero, Haydée Rotman, Regina Searle, Catherine
Beach, Isabel
Contreras-Peruyero, Haydée
Rotman, Regina
Searle, Catherine
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2025-06-11
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Article
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Keywords
Curvature bounds,Geodesic segments,Geodesics,Length bounds,Riemannian geometry,Sectional curvature
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Citation
Beach, I., Contreras-Peruyero, H., Rotman, R. et al. Linear Bounds for the Lengths of Geodesics on Manifolds with Curvature Bounded Below. J Geom Anal 35, 217 (2025). https://doi.org/10.1007/s12220-025-02003-6
Abstract
Let M be a simply connected Riemannian manifold in Mk,vD(n), the space of closed Riemannian manifolds of dimension n with sectional curvature bounded below by k, volume bounded below by v, and diameter bounded above by D. Let c be the smallest positive real number such that any closed curve of length at most 2d can be contracted to a point over curves of length at most cd, where d is the diameter of M. In this paper, we show that under these hypotheses there exists a computable rational function, G(n, k, v, D), such that any continuous map of Sl to Ωp,qM, the space of piecewise differentiable curves on M connecting p and q, is homotopic to a map whose image consists of curves of length at most exp(cexp(G(n,k,v,D))). In particular, for any points p,q∈M and any integer m>0, there exist at least m geodesics connecting p and q of length at most mexp(cexp(G(n,k,v,D))). © Mathematica Josephina, Inc. 2025.
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This is an open access article under the CC BY license.
Publisher
Springer
Journal
Journal of Geometric Analysis
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ISSN
10506926