On \(G_ 2\)-manifolds
type: Item
instance of: scholarly article
title: On \(G_ 2\)-manifolds
zbMATH Open document ID: 0804.53047
author: A. Kobotis
author: Philippos J. Xenos
published in: Annales Mathématiques Blaise Pascal
publication date: 1995-01-12T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/79096
review text: Let \((M,J,g)\) be a \((2m)\)-dimensional almost Hermitian manifold. Let \(\nabla\) denote the natural metric connection on \(M\) and \(R(X,Y)Z\) the curvature operator associated with \(\nabla\). The fundamental 2-form of \(M\) is \(F(X,Y)= g(\overline X, Y)\) for all vector fields \(X\) and \(Y\), where \(\overline X= JX\). An almost Hermitian manifold \((M,J,g)\) is said to be a \(G_ 2\)-manifold if and only if, for all vector fields \(X\), \(Y\) and \(Z\) on \(M\), holds \[ \underset{X,Y,Z} \sigma(\nabla_ X F)(Y,Z)= \underset{X,Y,Z} \sigma(\nabla_{\overline X} F)(\overline Y,Z), \] where by \(\sigma\) denote the cyclic sum [see also \textit{L. Hervella} and \textit{E. Vidal}, Nouvelles géométries pseudo-kählériennes \(G_ 1\) et \(G_ 2\), C. R. Acad. Sci., Paris, Ser. A 283, 115-118 (1976; Zbl 0331.53026)]. The authors study several properties of a \(G_ 2\)-manifold concerning with the curvature identities, the Chern classes and some results of type ``Schur's theorem''.
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C10
zbMATH DE Number: 605488
zbMATH Keywords: Schur's theorem
zbMATH Keywords: almost Hermitian manifold
zbMATH Keywords: curvature identities
zbMATH Keywords: Chern classes
reviewed by: Alberto Cavicchioli
MaRDI profile type: Publication
OpenAlex ID: W2023882519
cites work: Almost Hermitian manifolds with constant holomorphic sectional curvature
cites work: Q4097866
cites work: Q5565587
cites work: Some almost Hermitian manifolds with constant holomorphic sectional curvature
DOI: 10.5802/AMBP.3
description: scientific article; zbMATH DE number 605488