Linear Bounds for the Lengths of Geodesics on Manifolds With Curvature Bounded Below

2024/10/14 by Isabel Beach, Haydeé Contreras Peruyero, Beach, Isabel +5 · 1 citation
Computer Science · Mathematics · #53C22 #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2410.10975

openalex publication_date 2024/10/14 · openalex created_date 2024/10/20 · openalex updated_date 2026/07/28

Abstract

Let M be a simply connected Riemannian manifold in \mathscrM_k,v^D(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 S^l to Ω_p,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)).

Cited by

• Lengths of Orthogonal Geodesic Chords on Riemannian Manifolds

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