The Lie algebra of isometries of homogeneous spaces of nonpositive curvature
type: Item
instance of: scholarly article
title: The Lie algebra of isometries of homogeneous spaces of nonpositive curvature
zbMATH Open document ID: 0841.53045
DOI: 10.1007/BF01263526
author: Maria J. Druetta
published in: Geometriae Dedicata
publication date: 1996-07-14T00:00:00Z
review text: Let \(M\) denote a complete, simply connected homogeneous Riemannian manifold with sectional curvature \(K \leq 0\). Let \(G\) denote \(I_0(M)\), the identity component of the isometry group of \(M\), and let \(\mathfrak S\) denote the Lie algebra of \(G\). In this article the author finds sufficient conditions for \(M\) to be a symmetric space in terms of properties of \(G\) or \(\mathfrak G\). It is known that there exist simply transitive solvable subgroups of \(G\). Let \(S\) be such a group, and let \(\mathfrak G\) denote its Lie algebra. Then \(M\) is isometric to \(S\) with a left invariant metric \(\langle, \rangle\) and \(\{S, \langle, \rangle\}\) in turn is determined by the metric Lie algebra \(\{{\mathfrak S}, \langle, \rangle\}\) since \(S\) is simply connected and \({\mathfrak S} = T_eS\). Let \(K\) be a maximal compact subgroup of \(G\), and let \(\mathfrak K\) denote its Lie algebra. The Lie algebra \(\mathfrak S\) is said to be in standard position if \(B({\mathfrak K}, {\mathfrak A}) = 0\), where \(B\) denotes the Killing form of \(\mathfrak G\). It is always possible to find a simply transitive solvable group \(S\) whose Lie algebra is in standard position. Let \({\mathfrak A} = [{\mathfrak S}, {\mathfrak S}]^\perp\), the orthogonal complement in \(\mathfrak S\) of the commutator ideal \({\mathfrak S}' = [{\mathfrak S}, {\mathfrak S}]\). By work of Azencott and Wilson it is known that \(\mathfrak A\) is an abelian subalgebra of \(\mathfrak S\). Typically one now studies the geometry of \(M\) through the algebraic properties of the action of \(\text{ad}({\mathfrak A})\) on the complexifications \({\mathfrak G}^\mathbb{C}\) and \({\mathfrak S}^\mathbb{C}\) of \(\mathfrak G\) and \(\mathfrak S\), respectively. For each element \(\lambda \in\Hom({\mathfrak A},\mathbb{C})\) let \({\mathfrak G}^\mathbb{C}_n\) (respectively \({\mathfrak S}^\mathbb{C}_n)\) denote the largest subspace of \({\mathfrak G}^\mathbb{C}\) (respectively \({\mathfrak S}^\mathbb{C}…
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 806202
zbMATH Keywords: nonpositive curvature
zbMATH Keywords: symmetric space
zbMATH Keywords: solvable group
zbMATH Keywords: homogeneous space
reviewed by: Patrick Eberlein
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01263526
OpenAlex ID: W2075262505
description: scientific article; zbMATH DE number 806202