Structure of flat subspaces in low-dimensional manifolds of nonpositive curvature
type: Item
instance of: scholarly article
title: Structure of flat subspaces in low-dimensional manifolds of nonpositive curvature
zbMATH Open document ID: 0678.53035
DOI: 10.1007/BF01182086
published in: Manuscripta Mathematica
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155397
review text: The author determines the configuration of flat, totally geodesic submanifolds in a compact, real analytic manifold M of sectional curvature \(K\leq 0\) and dimension 3 or 4. This configuration depends only on the fundamental group of M (Corollary 2 below). Let \(\tilde M\) denote a complete, simply connected, real analytic manifold with sectional curvature \(K\leq 0\), and let \(M=\tilde M/\Gamma\) denote a compact quotient manifold of M, where \(\Gamma\) \(\subseteq I(\tilde M)\) is a discrete group isomorphic to the fundamental group of M. A k-flat \(F\subseteq \tilde M\) is a complete, flat, imbedded, totally geodesic submanifold of \(\tilde M.\) A complete, totally geodesic submanifold X of \(\tilde M\) is said to be closed (relative to M) if the image of X under the projection \(\pi: \tilde M\to M\) is compact. The following two questions are basic: 1) If \(\tilde M\) admits a k-flat F does it also admit a (possibly different) closed k-flat \(F'?\) 2) Are the closed k-flats of \(\tilde M\) dense in the space of all k-flats of \(\tilde M\)? These questions are unresolved for \(C^{\infty}\) manifolds of arbitrary dimension. The author gives an affirmative answer to both questions for real analytic manifolds of dimension 3 or 4. Theorem 1. Let \(M=\tilde M/\Gamma\) be a compact 3-dimensional real analytic manifold with sectional curvature \(K\leq 0\). If \(\tilde M\) is not a product \(\tilde M'\times R\), then there are (modulo \(\Gamma)\) only finitely many 2-flats in \(\tilde M.\) Each 2-flat is closed and any two different 2-flats \(F,F'\) in \(\tilde M\) are disjoint. A nonpositively curved manifold M is said to be of higher rank if every geodesic of the universal cover \(\tilde M\) is contained in a k-flat for some integer \(k\geq 2\). Theorem 2. Let \(M=\tilde M/\Gamma\) be a compact 4- dimensional real analytic manifold with sectional curvature \(K\leq 0\). Then there are (modulo \(\Gamma)\) only finitely many totally geodesic submanifolds of higher rank …
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 4111494
zbMATH Keywords: nonpositive curvature
zbMATH Keywords: low dimensions
zbMATH Keywords: totally geodesic submanifolds
zbMATH Keywords: real analytic manifold
zbMATH Keywords: fundamental group
zbMATH Keywords: k-flat
author: Viktor Schroeder
reviewed by: Patrick Eberlein
MaRDI profile type: Publication
cites work: Existence of flats in manifolds of nonpositive curvature
cites work: Nonpositively curved manifolds of higher rank
cites work: Structure of manifolds of nonpositive curvature. I
cites work: Fundamental groups of manifolds of nonpositive curvature
cites work: Manifolds of nonpositive curvature
cites work: Manifolds of nonpositive curvature and their buildings
cites work: Geodesic Flow in Certain Manifolds Without Conjugate Points
cites work: Euclidean de Rham factor of a lattice of nonpositive curvature
cites work: Manifolds of negative curvature
cites work: Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature
cites work: Stratification of real analytic mappings and images
cites work: Compact manifolds of nonpositive curvature
cites work: Existence of immersed tori in manifolds of nonpositive curvature.
cites work: Rigidity of nonpositively curved graph manifolds
description: scientific article; zbMATH DE number 4111494