Rigidity for closed manifolds with positive curvature
type: Item
instance of: scholarly article
title: Rigidity for closed manifolds with positive curvature
zbMATH Open document ID: 1178.53033
published in: Annals of Global Analysis and Geometry
publication date: 2009-08-12T00:00:00Z
review text: Let \(M\) be an \(n\)-dimensional Riemannian manifold with sectional curvature \(\mathrm{sec}(M)\geq 1\), let \(d\) be the distance function and \(\mathrm{rad}(M)\) be the radius of \(M\). Wang proved that if \(\mathrm{rad}(M)> \pi/2\) then the length of any closed geodesic of \(M\) is bigger or equal to \(2\cdot \mathrm{rad}(M)\). In this paper, rigidity of positively curved manifolds with large radius is studied by the existence of special geodesics. Let \(\mathrm{conj}(x)\), the conjugate locus of \(M\) at \(x\), be the set of the first conjugate points to \(x\), for all geodesics starting at \(x\). Denote by \(S_xM\) the unit tangent sphere of \(M\) at \(x\). For any \(v\in S_xM\), the conjugate value \(c_v\) is the first number \(r>0\) such that thre is a nonzero Jacobi field \(J\), along the geodesic from \(x\) in the direction of \(v\), satisfying \(J(0)=J(r)=0\). It is shown that any \(n\)-dimensional complete connected Riemannian manifold \(M\) with \(\mathrm{sec}(M)\geq 1\) and \(\mathrm{rad}(M)>\pi/2\), such that for any \(x\in M\), the first conjugate locus of \(x\) is a single point and such that \(M\) contains a geodesic loop of length \(2\cdot\mathrm{rad}(M)\), is isometric to a round \(n\)-sphere. Further, it is shown that any \(n\)-dimensional complete connected Riemannian manifold \(M\) with \(\mathrm{sec}(M)\geq 1\) and \(\mathrm{rad}(M)>\pi/2\), such that for any \(x\in M\) the first conjugate value function \(c_x: S_xM\rightarrow {\mathbb R}\) is a constant function, is isometric to a round \(n\)-sphere.
reviewed by: Zdeněk Dušek
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 5594049
zbMATH Keywords: closed manifolds
zbMATH Keywords: conjugate locus
zbMATH Keywords: rigidity
author: Changyu Xia
MaRDI profile type: Publication
full work available at URL: s10455-008-9151-7
OpenAlex ID: W1973048922
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DOI: 10.1007/S10455-008-9151-7
description: scientific article; zbMATH DE number 5594049