Manifolds all of whose flats are closed
type: Item
instance of: scholarly article
title: Manifolds all of whose flats are closed
zbMATH Open document ID: 0901.53030
author: Bernardo Molina
author: Carlos Olmos
published in: Journal of Differential Geometry
publication date: 1998-11-22T00:00:00Z
review text: A compact Riemannian manifold is of higher rank if any geodesic lies in a flat totally geodesic submanifold (``flat'') of dimension \(\geq 2\). If such a manifold \(M\) has sectional curvature \(K \leq 0\), then the well-known rank rigidity theorem mainly due to W. Ballmann says that \(M\) is locally symmetric. Without the curvature assumption, this is wrong; in particular, it fails for \(K \geq 0\): the easiest counterexample is a normally homogeneous space \(G/T'\) where \(G\) is a compact Lie group and \(T'\) a subtorus of a maximal torus with codimension \(\geq 2\). However, in the counterexamples almost all flats are noncompact. In the present paper, the following main theorems are shown (in fact, the assumptions made are slightly more general). Theorem If \(M\) is compact and locally irreducible such that any geodesic lies in a compact flat of dimension \(\geq 2\), then \(M\) is locally symmetric. Corollary. A compact simply connected irreducible analytic manifold all of whose geodesics lie in a closed flat is either symmetric or a manifold all of whose geodesics are closed. The central idea of the proof is as follows. By the holonomy theorem of Berger and Simons it suffices to show that the local holonomy group at any point \(p\) in some open dense subset does not act transitively on the unit sphere of \(T_pM\). Assume the contrary and consider a parallel unit vector field \(v(t)\) along a short piecewise smooth closed path \(c(t)\) starting and ending at \(p\). One finds a flat \(k\)-torus \(T\) (with \(k \geq 2\)) and a piecewise smooth family of totally geodesic affine immersions \(f_t : T \to M\) passing through \(c(t)\) and being tangential to \(v(t)\). Thus \({\partial \over \partial t}f_t\) is a Jacobi field along \(f_t\). As in the case where \(f_t\) is a geodesic, this splits into a tangent and a normal component with respect to \(f_t\), where the tangent component is affine and hence parallel by periodicity. Let \(v_0\) be the tangent vector of \(T\…
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 1145704
zbMATH Keywords: local holonomy
zbMATH Keywords: rank rigidity
zbMATH Keywords: immersed tori
zbMATH Keywords: Baire category
Wikidata QID: Q115175728
reviewed by: J.-H. Eschenburg
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214459843
OpenAlex ID: W1567218081
DOI: 10.4310/JDG/1214459843
description: scientific article; zbMATH DE number 1145704