On some compact almost Kähler locally symmetric space
type: Item
instance of: scholarly article
title: On some compact almost Kähler locally symmetric space
zbMATH Open document ID: 0895.53046
DOI: 10.1155/S0161171298000076
author: Takashi Oguro
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1998-03-11T00:00:00Z
full work available at URL: https://eudml.org/doc/47956
review text: An almost Hermitian manifold \((M^{2n},g,J)\) is said to be almost Kählerian if its Kähler form \(\Omega\) is closed. S. Goldberg conjectured that a compact almost Kähler Einstein space is necessarily Kählerian, i.e., the almost complex structure is integrable. K. Sekigawa showed that this conjecture holds for non-negative scalar curvature. In this paper the author considers a \(*\)-version of this result and proves: a compact almost Kähler weakly \(*\)-Einstein space which is locally symmetric and has non-negative \(*\)-scalar curvature is Kählerian. \((M,g,J)\) is said to be a weakly \(*\)-Einstein space if the Ricci \(*\)-tensor \(\rho^*\) satisfies \(2n\rho^* = \tau^*g\) where \(\tau^*\) is the \(*\)-scalar curvature and \(\rho^*\) is defined by \(\rho^*(x,y)=\sum^{2n}_{i=1} R(x,e_i,Jy,Je_i)\) for an arbitrary orthonormal basis \(\{e_i, i=1,\dots,2n\}\) of the tangent space at each point.
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 1127741
zbMATH Keywords: compact almost Kähler manifold
zbMATH Keywords: locally symmetric manifold
zbMATH Keywords: weakly \(*\)-Einstein manifold
zbMATH Keywords: non-negative \(*\)-scalar curvature
reviewed by: Lieven Vanhecke
MaRDI profile type: Publication
OpenAlex ID: W2011579675
description: scientific article; zbMATH DE number 1127741