Compact almost Kähler manifolds with divergence-free Weyl conformal tensor
type: Item
instance of: scholarly article
title: Compact almost Kähler manifolds with divergence-free Weyl conformal tensor
zbMATH Open document ID: 1080.53068
author: Hiroyasu Satoh
published in: Annals of Global Analysis and Geometry
publication date: 2005-04-07T00:00:00Z
review text: Let \((M^{2n},J,g)\) be an almost Hermitian manifold, where \(J\) is an almost complex structure, and \(g\) is a \(J\)-invariant metric on \(M\). \((M,J,g)\) is an almost Kähler manifold if the Kähler form is closed. Kähler manifolds are almost Kähler manifolds. However, in general the converse is not true. Let \(W\) be the Weyl conformal tensor, \(\delta W(Z,X,Y)=-\text{Trace}_g\nabla W(\cdot,Z,X,Y)\), where \(\nabla\) is the Levi-Civita connection of \(g\). \(J\) acts on the bundle of 2-forms \(\Lambda^2\) as an involutive endomorphism. Let \(\Lambda^{J+}\), \(\Lambda^{J-}\) be eigen-subbundles of eigenvalues \(\pm1\), and let \(\pi_2:\,\Lambda^1\otimes\Lambda^2\rightarrow\Lambda^1\otimes\Lambda^{J-}\) be the projection. The author proves that \((M,J,g)\) is a Kähler manifold assuming that \((M,J,g)\) is a compact \(2n\)-dimensional almost Kähler manifold whose Ricci tensor is positive semi-definite, and whose Weyl conformal tensor \(W\) satisfies the equation \(\pi_2(\delta W)=0\).
reviewed by: Neda Bokan
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 2153680
zbMATH Keywords: Goldberg conjecture
zbMATH Keywords: Kähler form
zbMATH Keywords: Pontryagin class
MaRDI profile type: Publication
Wikidata QID: Q125927319
full work available at URL: b:agag.0000031159.07314.8f
OpenAlex ID: W2000367291
DOI: 10.1023/B:AGAG.0000031159.07314.8F
description: scientific article; zbMATH DE number 2153680