On almost hyper-para-Kähler manifolds
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 \(K\).
scientific article; zbMATH DE number 6031943
scientific article