On Almost Hyper-Para-Kähler Manifolds

Abstract

In this paper it is shown that a -dimensional almost symplectic manifold can be endowed with an almost paracomplex structure , , and an almost complex structure , , satisfying for , for and , if and only if the structure group of can be reduced from (or ) to . In the symplectic case such a manifold is called an almost hyper-para-Kähler manifold. Topological and metric properties of almost hyper-para-Kähler manifolds as well as integrability of are discussed. It is especially shown that the Pontrjagin classes of the eigenbundles of to the eigenvalues depend only on the symplectic structure and not on the choice of .