Splitting theorems for nonnegatively curved open manifolds with large ideal boundary
type: Item
instance of: scholarly article
title: Splitting theorems for nonnegatively curved open manifolds with large ideal boundary
zbMATH Open document ID: 0791.53045
DOI: 10.1007/BF02571655
author: Takashi Shioya
published in: Mathematische Zeitschrift
publication date: 1994-04-12T00:00:00Z
full work available at URL: https://eudml.org/doc/174484
review text: Gromov defined the ideal boundary \(M(\infty)\) with the Tits metric \(d_ \infty\) of a nonnegatively curved complete open Riemannian manifold \(M\). It is our purpose to study the relation between the ideal boundary \(M(\infty)\) and the structure of \(M\). We consider the radius of the ideal boundary, which is defined by \(\text{rad } M(\infty):= \inf_{y\in M(\infty)} \sup_{x\in M(\infty)} d_ \infty (x,y)\), and prove the following statement. For any integer \(n\geq 2\) there exists an \(\varepsilon(n)>0\) such that if a nonnegatively curved \(n\)-dimensional complete open Riemannian manifold \(M\) satisfies \(\text{rad } M(\infty)>\pi- \varepsilon(n)\), then \(M\) is isometric to the Riemannian product \(S\times N\), where \(S\) is a compact Riemannian manifold and \(N\) a complete Riemannian manifold diffeomorphic to \(\mathbb{R}^ k\). Note here that \(N\) is not necessarily isometric to \(\mathbb{R}^ k\) and that \(\text{rad } M(\infty)\) is not equal to \(\pi\) in general.
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 549730
zbMATH Keywords: ideal boundary
zbMATH Keywords: Tits metric
zbMATH Keywords: Riemannian product
MaRDI profile type: Publication
cites work: Manifolds of nonpositive curvature
cites work: On the structure of complete manifolds of nonnegative curvature
cites work: Collapsing Riemannian manifolds to ones of lower dimensions
cites work: Convex functions on complete noncompact manifolds: Topological structure
cites work: Convex functions on complete noncompact manifolds : differentiable structure
cites work: Q5644742
cites work: Volume comparison à la Aleksandrov
cites work: A compactification of a manifold with asymptotically nonnegative curvature
cites work: On Manifolds of Small Excess
cites work: Q3473766
cites work: Q3977566
cites work: Nonnegatively curved manifolds with souls of codimension 2
cites work: A Splitting Theorem for 4-Dimensional Manifolds of Nonnegative Curvature
cites work: Space of souls in a complete open manifold of nonnegative curvature
Recommended article: A splitting theorem for open nonnegatively curved manifolds
Recommended article: A Splitting Theorem for Complete Manifolds with Nonnegative Curvature Operator
Recommended article: A splitting theorem for spaces of nonpositive curvature
Recommended article: Splitting theorems on complete Riemannian manifolds with nonnegative Ricci curvature
Recommended article: A splitting theorem for manifolds of almost nonnegative Ricci curvature
Recommended article: A splitting theorem for spaces of Busemann non-positive curvature
Recommended article: Splitting theorems for submanifolds of nonnegative isotropic curvature
Recommended article: A splitting theorem for Euclidean submanifolds of nonpositive sectional curvature
Recommended article: Splitting theorems for Finsler manifolds of nonnegative Ricci curvature
Recommended article: A splitting theorem for nonnegatively curved Alexandrov spaces
description: scientific article; zbMATH DE number 549730