On almost hyper-para-Kähler manifolds
type: Item
instance of: scholarly article
title: On almost hyper-para-Kähler manifolds
zbMATH Open document ID: 1239.53039
DOI: 10.5402/2012/535101
author: Jochen Merker
published in: ISRN Geometry
publication date: 2012-05-09T00:00:00Z
review text: Summary: In this paper it is shown that a \(2n\)-dimensional almost symplectic manifold \((M, \omega)\) can be endowed with an almost paracomplex structure \(K, K^2 = \text{Id}_{TM}\), and an almost complex structure \(J, J^2 = -\text{Id}_{TM}\), satisfying \(\omega(JX, JY) = \omega(X, Y) = -\omega(KX, KY)\) for \(X, Y \in TM, \omega(X, JX) > 0\) for \(X \neq 0\) and \(KJ = -JK\), if and only if the structure group of \(TM\) can be reduced from \(Sp(2n)\) (or \(U(n)\)) to \(O(n)\). In the symplectic case such a manifold \((M, \omega, J, K)\) is called an almost hyper-para-Kähler manifold. Topological and metric properties of almost hyper-para-Kähler manifolds as well as integrability of \((J, K)\) are discussed. It is especially shown that the Pontrjagin classes of the eigenbundles \(P_\pm\) of \(K\) to the eigenvalues \(\pm 1\) depend only on the symplectic structure and not on the choice of \(K\).
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53D05
Mathematics Subject Classification ID: 32Q15
zbMATH DE Number: 6031943
zbMATH Keywords: almost symplectic
zbMATH Keywords: almost paracomplex structure
zbMATH Keywords: integrability
zbMATH Keywords: Pontrjagin classes
Wikidata QID: Q58691182
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.5402/2012/535101
OpenAlex ID: W1989651438
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description: scientific article; zbMATH DE number 6031943