On Almost Hyper-Para-Kähler Manifolds. (8th March 2012)

• Record Type: Journal Article
• Authors: Merker, Jochen
• Other Names: Aluffi P. Academic Editor.; Rosenbaum M. Academic Editor.
• Is Part Of: ISRN geometry. Volume 2012(2012)
• Publication Date: 2012-03-08
• Subjects: Geometry -- Periodicals; Topology -- Periodicals; Geometry; Topology; Periodicals; 516
• DOI: 10.5402/2012/535101
• Languages: English
• ISSNs: 2090-6307

Abstract

Abstract : In this paper it is shown that a 2 n -dimensional almost symplectic manifold ( M, ω ) can be endowed with an almost paracomplex structure K, K 2 = I d T M, and an almost complex structure J, J 2 = - I d T M, satisfying ω ( J X, J Y ) = ω ( X, Y ) = - ω ( K X, K Y ) for X, Y ∈ T M, ω ( X, J X ) > 0 for X ≠ 0 and K J = - J K, if and only if the structure group of T M can be reduced from S p ( 2 n ) (or U ( n ) ) to O ( n ) . In the symplectic case such a manifold ( M, ω, 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 ± of K to the eigenvalues ± 1 depend only on the symplectic structure and not on the choice of K .