A geometric characterization of negatively curved locally symmetric spaces
type: Item
instance of: scholarly article
title: A geometric characterization of negatively curved locally symmetric spaces
zbMATH Open document ID: 0733.53018
author: Ursula Hamenstädt
published in: Journal of Differential Geometry
publication date: 1991-01-01T00:00:00Z
review text: The author proves a rank 1 version of an earlier higher rank rigidity result obtained independently by W. Ballmann and K. Burns - R. Spatzier. Theorem. Let M be a compact Riemannian manifold with sectional curvature \(K\leq -1\). Assume that for each unit vector \(v\in T_ 1\tilde M\) there is a parallel, perpendicular vector field E(t) on the corresponding geodesic \(\gamma_ V\) such that \(K(E,\gamma_ v')(t)=-1\) for all \(t\in {\mathbb{R}}\). Then M is locally symmetric of rank 1. The rigidity result of Ballmann and Burns-Spatzier can be formulated in a similar way by replacing -1 by zero in the statement above. In the opinion of the reviewer these two results should be regarded as two halves of one more general result although the proofs of the two results are quite different. The common link is the rigidity caused by the presence of parallel, perpendicular vector fields along each geodesic that determine 2-planes with maximum sectional curvatures. Essentially the same dichotomy arises in the Mostow Rigidity Theorem, where the proof falls into two cases: a rank 1 case involving quasiconformal analysis and a higher rank case involving Tits buildings. We outline some ideas of the proof of the author's theorem. Let M denote the universal cover of M and let \(\tilde M(\infty)\) denote the boundary sphere of \(\tilde M.\) 1) For each vector \(v\in T_ 1\tilde M\) the author introduces a non- Riemannian metric \(\eta_ v\) on \(M(\infty)-\{\gamma_ v(-\infty)\}\), a set which can be identified naturally with the strong unstable horosphere \(W^{su}(v)\subseteq T_ 1M\) determined by v. The metrics \(\eta_ v\) depend on v but are 1-quasiconformally equivalent on \(\tilde M(\infty).\) 2) For each \(v\in T_ 1M\) and each \(\xi \in T_ v(T_ 1\tilde M)\), the vector field \(Y_{\xi}(t)=(d\pi \circ g^ t)(\xi)\) is a Jacobi vector field on \(\gamma_ v\), where \(\{g^ t\}\) denotes the geodesic flow in \(T_ 1\tilde M\) and \(\pi: T_ 1\tilde M\to \tilde M\) is the projection. Let \(E_…
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 4212923
zbMATH Keywords: rank 1
zbMATH Keywords: rigidity result
zbMATH Keywords: sectional curvature
zbMATH Keywords: locally symmetric
zbMATH Keywords: boundary sphere
zbMATH Keywords: geodesic flow
zbMATH Keywords: Carnot-Caratheodory metric
zbMATH Keywords: 1-quasiconformal structure
Wikidata QID: Q115181537
reviewed by: Patrick Eberlein
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214446998
OpenAlex ID: W1492194752
DOI: 10.4310/JDG/1214446998
description: scientific article; zbMATH DE number 4212923