Negatively curved homogeneous Osserman spaces
type: Item
instance of: scholarly article
title: Negatively curved homogeneous Osserman spaces
zbMATH Open document ID: 0970.53031
DOI: 10.1016/S0926-2245(99)00032-7
author: Maria J. Druetta
author: I. G. Dotti
published in: Differential Geometry and its Applications
publication date: 2001-10-14T00:00:00Z
review text: Let \(M\) be a riemannian manifold, \(R\) its curvature tensor and \(R_X\) the curvature operator, defined by \(R_{X}Y = R(Y,X)X\), \(X\) a unit tangent vector. When \(R_X\) has constant eigenvalues, counting multiplicities, independently of \(X\) in \(T_{p}M\) and of the choice of \(p\) in \(M\), the riemannian manifold is said to satisfy the Osserman condition. It is immediate to see that it is satisfied by rank one symmetric spaces. Based on results obtained in R. Osserman, P. Sarnak, A new curvature invariant and entropy of geodesic flows, Inventiones Math, 77 (1984) 455-462, Osserman cojectured that locally rank one symmetric spaces are the only non-flat riemannian manifolds satisfying this condition. The authors proved the validity of the above conjecture in the case of a homogeneous manifold of nonpositive curvature. An outline of the proof is as follows. Every riemannian manifold satisfying the Osserman condition is Einstein, hence by applying results from (J. Heber, Noncompact homogeneous Einstein spaces, Inventiones Math. 133 (1998) 279-352) it is sufficient to verify the conjecture in the case of a solvable Lie groups of Iwasawa type. This is proved in Theorem 3.4: If \(\mathcal S\) is a Lie algebra of Iwasawa type satisfying the Osserman condition, then the associated simply connected Lie group \(S\) with left invariant metric is a symmetric space of noncompact type and rank one.
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 1367713
zbMATH Keywords: Osserman condition
zbMATH Keywords: solvable Lie group
zbMATH Keywords: Iwasawa type
zbMATH Keywords: curvature operator
zbMATH Keywords: homogeneous manifold
zbMATH Keywords: nonpositive curvature
Wikidata QID: Q115337406
reviewed by: Andrzej Szczepański
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1367713