On Almost Hyper-Para-Kähler Manifolds

Abstract EN

In this paper it is shown that a 2n-dimensional almost symplectic manifold (M,ω) can be endowed with an almost paracomplex structure K, K2=IdTM, and an almost complex structure J, J2=-IdTM, satisfying ω(JX,JY)=ω(X,Y)=-ω(KX,KY) for X,Y∈TM, ω(X,JX)>0 for X≠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,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.