Naturally reductive locally 4-symmetric spaces
type: Item
instance of: scholarly article
title: Naturally reductive locally 4-symmetric spaces
zbMATH Open document ID: 0656.53050
DOI: 10.1007/BF02765040
author: Arthur J. Ledger
published in: Israel Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: The class of naturally reductive pseudo-Riemannian 3-symmetric spaces has been determined by \textit{A. Gray} as that of all nearly Kaehler manifolds satisfying an additional (simple) condition for the covariant derivative of the Riemannian curvature [cf. J. Differ. Geom. 7, 343-369 (1972; Zbl 0275.53026)]. As an analogue, the present author characterizes all naturally reductive 4-symmetric spaces as (pseudo) Riemannian manifolds with an f-structure (in the sense of K. Yano) satisfying (i) three identities analogous to the nearly Kaehler property, and (ii) two additional (simple) conditions for the covariant derivative of the curvature. Notice that the number of conditions is larger because, besides the f-structure F satisfying \(F^ 3+F=0\), one has to consider also its associated polynomial structure \(P=F^ 2+I\). Problems of this kind (also for higher orders k) are interesting and far from being obvious.
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 4071844
zbMATH Keywords: k-symmetric space
zbMATH Keywords: naturally reductive 4-symmetric spaces
zbMATH Keywords: f-structure
zbMATH Keywords: polynomial structure
reviewed by: Oldřich Kowalski
MaRDI profile type: Publication
cites work: Q5654724
cites work: Q5667207
cites work: Nearly Kähler manifolds
cites work: Riemannian manifolds with geodesic symmetries of order 3
cites work: Riemannian 4-Symmetric Spaces
cites work: Q5731354
cites work: Generalized symmetric spaces
cites work: Q4112582
cites work: Q5525849
cites work: Q3942011
cites work: Affine and Riemannian s-manifolds
cites work: Compact quadratic s-manifolds
cites work: Homogeneous spaces defined by Lie group automorphisms. I
cites work: Homogeneous spaces defined by Lie group automorphisms. II
description: scientific article; zbMATH DE number 4071844