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September 2021 Critical values of homology classes of loops and positive curvature
Hans-Bert Rademacher
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J. Differential Geom. 119(1): 141-159 (September 2021). DOI: 10.4310/jdg/1631124316

Abstract

We study compact and simply-connected Riemannian manifolds $(M,g)$ with positive sectional curvature $K \geq 1$. For a nontrivial homology class of lowest positive dimension in the space of loops based at a point $p \in M$ or in the free loop space one can define a critical length $\mathsf{crl}_p (M,g)$ resp. $\mathsf{crl} (M,g)$. Then $\mathsf{crl}_p (M,g)$ equals the length of a geodesic loop with base point $p$ and $\mathsf{crl} (M,g)$ equals the length of a closed geodesic. This is the idea of the proof of the existence of a closed geodesic of positive length presented by Birkhoff in case of a sphere and by Lusternik & Fet in the general case. It is the main result of the paper that the numbers $\mathsf{crl}_p (M,g)$ resp. $\mathsf{crl} (M,g)$ attain its maximal value $2\pi$ only for the round metric on the $n$-sphere.

Under the additional assumption $K \leq 4$ this result for $\mathsf{crl} (M,g)$ follows from results by Sugimoto in even dimensions and Ballmann, Thorbergsson & Ziller in odd dimensions.

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Hans-Bert Rademacher. "Critical values of homology classes of loops and positive curvature." J. Differential Geom. 119 (1) 141 - 159, September 2021. https://doi.org/10.4310/jdg/1631124316

Information

Received: 27 October 2017; Accepted: 6 December 2019; Published: September 2021
First available in Project Euclid: 10 September 2021

Digital Object Identifier: 10.4310/jdg/1631124316

Subjects:
Primary: 53C20 , 53C21 , 53C22 , 53C24 , 58E10

Keywords: free loop space , geodesic loops , loop space , Morse theory , positive sectional curvature

Rights: Copyright © 2021 Lehigh University

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Vol.119 • No. 1 • September 2021
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