Quaternionic Kähler manifolds with Hermitian and Norden metrics

Abstract

Abstract. Almost hypercomplex manifolds with Hermitian and Norden metrics and more specially the corresponding quaternionic Kähler manifolds are considered. Some necessary and sufficient conditions the investigated manifolds be isotropic hyper-Kählerian and flat are found. It is proved that the quaternionic Kähler manifolds with the considered metric structure are Einstein for dimension at least 8. The class of the non-hyper-Kähler quaternionic Kähler manifold of the considered type is determined. ContentsIntroduction 1 1. Almost hypercomplex manifolds with NH-metric structure 2 2. Quaternionic Kähler manifolds with NH-metric structure 4 3. Quaternionic Kähler NH-manifolds in a classification of almost hypercomplex NH-manifolds 7 4. Non-hyper-Kähler quaternionic Kähler NH-manifolds 9 References 10 IntroductionIn this work * we continue the investigations on a manifold M with an almost hypercomplex structure H. We equip this almost hypercomplex manifold (M, H) with a metric structure G, generated by a pseudo-Riemannian metric g of neutral signature ([4], [5]).It is known, if g is a Hermitian metric on (M, H), the derived metric structure G is the known hyper-Hermitian structure. It consists of the given Hermitian metric g with respect to the three almost complex structures of H and the three Kähler forms associated with g by H [1].In our case the considered metric structure G has a different type of compatibility with H. The structure G is generated by a neutral metric g such that the one (resp., the other two) of the almost complex structures of H acts as an isometry (resp., act as anti-isometries) with respect to g in each tangent fibre. Let the almost complex structures of H act as isometries or anti-isometries with respect to the metric. Then the existence of an anti-isometry generates exactly the existence of one more antiisometry and an isometry. Thus, G contains the metric g and three (0,2)-tensors associated by H -a Kähler form and two metrics of the same type. The existence of such bilinear forms on an almost hypercomplex manifold is proved in [4]. The neutral metric g is Hermitian with respect to the one almost complex structure of H and g is an anti-Hermitian (i. e. a Norden) metric regarding the other two almost complex structures of H. For this reason we call the derived almost hypercomplex manifold (M, H, G) an almost hypercomplex manifold with NH-metric structure or an almost hypercomplex NH-manifold.If the three almost complex structures of H are parallel with respect to the LeviCivita connection ∇ of g then such hypercomplex NH-manifolds of Kähler type we call hyper-Kähler NH-manifolds, which are flat according to [5].In the first section we recall some facts about the almost hypercomplex NHmanifolds known from [1], [4], [5], [7].In the second section we introduce the corresponding quaternionic Kähler manifold of an almost hypercomplex manifold with NH-metric structure. We establish that the quaternionic Kähler NH-manifolds are Einstein for dimension 4n ≥ 8. For comparison, it is known that the ...