Nonpositively curved manifolds of higher rank
type: Item
instance of: scholarly article
title: Nonpositively curved manifolds of higher rank
zbMATH Open document ID: 0585.53031
DOI: 10.2307/1971331
author: Werner Ballmann
published in: Annals of Mathematics. Second Series
publication date: 1985-01-01T00:00:00Z
review text: The author investigates the structure of the universal covering space \(\tilde M\) of a complete Riemannian manifold M with nonpositive sectional curvature bounded from below by some constant \(-a^ 2\). The rank of a unit tangent vector v of M is defined to be the dimension of the space of parallel Jacobi vector fields along the geodesic \(\gamma_ v\) with the initial velocity v. Further, the rank of the Riemannian manifold M is defined as \[ rank(M)=\min \{rank(v)| \quad v\in SM\}. \] The main theorem says that if M has finite volume, then \(\tilde M\) is a space of rank one, or a symmetric space, or a product of such spaces. Corollary: Suppoe that M has finite volume. If M is irreducible and \(rank(M)=k\geq 2\), then M is a locally symmetric space of noncompact type of rank k.
reviewed by: Oldřich Kowalski
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 3937987
zbMATH Keywords: nonpositive sectional curvature
zbMATH Keywords: Jacobi vector fields
zbMATH Keywords: space of rank one
zbMATH Keywords: finite volume
zbMATH Keywords: locally symmetric space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971331
OpenAlex ID: W2019788826
description: scientific article; zbMATH DE number 3937987