Isotropic Kähler structures on Engel 4-manifolds
type: Item
instance of: scholarly article
title: Isotropic Kähler structures on Engel 4-manifolds
zbMATH Open document ID: 0964.53047
DOI: 10.1016/S0393-0440(99)00055-8
author: Eduardo García-Río
author: Yasuo Matsushita
published in: Journal of Geometry and Physics
publication date: 2001-07-16T00:00:00Z
review text: The authors of this paper consider indefinite almost Hermitian manifolds \((M,g,J)\) whose almost complex structure \(J\) satisfies the following two conditions \(\nabla J\neq 0\) and \(\|\nabla J\|^2= 0\), \(\nabla\) being the Levi-Civita connection of the metric \(g\). When the second condition holds \(M\) is called an isotropic Kähler manifold. It is to remark that the second condition does not imply the Kähler condition \(\nabla J= 0\), i.e., the isotropic Kähler manifolds are not indefinite Kähler. Examples of non-Kähler isotropic Kähler manifolds are constructed in the case \(M= \mathbb{R}^4\) with an Engel structure (= a special absolutely non-integrable regular two-dimensional distribution). It is shown that \(M\) is endowed with two kinds of indefinite almost Hermitian structures \((g, J)\) and \((g,J')\) which are isotropic Kähler but neither complex nor symplectic. Isotropic Kähler almost Kähler manifolds are defined for which all linear invariants vanish.
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 1437068
zbMATH Keywords: indefinite Kähler manifold
zbMATH Keywords: Engel distribution
zbMATH Keywords: almost complex structure
zbMATH Keywords: isotropic Kähler manifold
zbMATH Keywords: almost Hermitian structure
reviewed by: Stancho Dimiev
MaRDI profile type: Publication
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full work available at URL: s0393-0440(99)00055-8
OpenAlex ID: W2076242067
description: scientific article; zbMATH DE number 1437068