Table of Contents
ISRN Geometry
Volume 2012, Article ID 535101, 13 pages
http://dx.doi.org/10.5402/2012/535101
Research Article

On Almost Hyper-Para-KΓ€hler Manifolds

Institute of Mathematics, University of Rostock, Ulmenstr. 69 (Haus 3), 18057 Rostock, Germany

Received 28 November 2011; Accepted 27 December 2011

Academic Editors: P. Aluffi and M. Rosenbaum

Copyright © 2012 Jochen Merker. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper it is shown that a 2 𝑛 -dimensional almost symplectic manifold ( 𝑀 , πœ” ) can be endowed with an almost paracomplex structure 𝐾 , 𝐾 2 = I d 𝑇 𝑀 , and an almost complex structure 𝐽 , 𝐽 2 = βˆ’ I d 𝑇 𝑀 , satisfying πœ” ( 𝐽 𝑋 , 𝐽 π‘Œ ) = πœ” ( 𝑋 , π‘Œ ) = βˆ’ πœ” ( 𝐾 𝑋 , 𝐾 π‘Œ ) for 𝑋 , π‘Œ ∈ 𝑇 𝑀 , πœ” ( 𝑋 , 𝐽 𝑋 ) > 0 for 𝑋 β‰  0 and 𝐾 𝐽 = βˆ’ 𝐽 𝐾 , if and only if the structure group of 𝑇 𝑀 can be reduced from 𝑆 𝑝 ( 2 𝑛 ) (or π‘ˆ ( 𝑛 ) ) to 𝑂 ( 𝑛 ) . In the symplectic case such a manifold ( 𝑀 , πœ” , 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 Β± of 𝐾 to the eigenvalues Β± 1 depend only on the symplectic structure and not on the choice of 𝐾 .