Almost hypercomplex structure. f generalized almost complex structures. Definition 3. It is given a geometric interpretation of the vanishing of these tensors as a By comparison, the structures that we treat in this article arise from almost symplectic forms ω which are H-Hermitian, respectively Q-Hermitian. There are studied Lie groups considered as almost hypercomplex Hermitian-Norden manifolds, which are integrable and have the lowest dimen-sion four. Here g is a neutral metric, which is Hermitian with respect to the almost complex structure J1 of H and g is a Norden metric 1 INTRODUCTION Hypercomplex and quaternionic geometries of skew-Hermitian type are geometric structures induced by pairs , respectively , where is an almost hypercomplex structure, is an almost quaternionic structure, and is an -Hermitian, respectively -Hermitian, almost symplectic 2-form. In Section 5, we focus on hypercomplex almost abelian Lie groups. Such Lie groups were studied in a recent paper of Andrada and Barberis [1]. Theorem 2. The reduction of the structure group to GL (n, H) is possible if and only if the almost quaternionic structure bundle H ⊂ End (T M) is trivial (i. When the family consists of generalized complex structures, we obt in a generalized hypercomplex structure. In differential geometry, a hypercomplex manifold is a manifold with the tangent bundle equipped with an action by the algebra of quaternions in such a way that the quaternions define integrable almost complex structures. We obtain the classi cation of hypercomplex almost abelian Lie groups in dimen-sion 8 and determine which ones admit In the second we develop the notion of an almost hypercomplex manifold with a pseudo-Hermitian structure and particularly the so-called pseudo-hyper-K ̈ahler-ian and isotropic K ̈ahler structures. 1. If a homogeneous space G/H, where G is a compact Lie group and H is a closed connected subgroup, admits an invariant almost hypercomplex structure, then there are no non-zero complementary roots for G related to H. We give a self-contained proof of desingularization theorem for hy- percomplex varieties: a normalization of a hypercomplex variety is smooth and hypercomplex. Its torsion is defined by 〚 〛 and its curvature by 〚 〛 for all , where 〚 〛 is the Courant bracket of and . The corresponding intrinsic torsions were computed in the previous article in this series, and the algebraic types of the geometries were derived, together with the minimal adapted An associated Nijenhuis tensor of endomorphisms in the tangent bundle is introduced. Moreover, we determine which 12-dimensional simply con-nected hypercomplex almost abelian Lie groups admit lattices. We determine when such Lie groups admit HKT metrics and study the corresponding Bismut connection. We study almost hypercomplex structure with Hermitian–Norden metrics on 4-dimensional Lie groups considered as smooth manifolds. It is proved that the quater-nionic K ̈ahler manifolds with the considered metric structure are Einstein for Abstract. Now classical definition of hypercomplexity requires all of them to be integrable. We similarly obtain that on unimodular almost abelian Lie groups a left-invariant hyperhermitian structure is HKT if and only if it is hyperk ̈ahler. Finally, for each integer n > 1 we He also constructed an almost hypercomplex structure on the total space of the tangent bundle of any almost complex manifold. These manifolds are equipped with an almost hypercomplex structure H = (J1, J2, J3) and a metric structure G = (g, g1, g2, g3). Special attention is paid to such tensors for an almost hypercomplex structure and the metric of Hermitian Moreover, we define an almost hypercomplex structure ( J ¯ 1 , J ¯ 2 , J ¯ 3 ) on the cotangent bundle T ∗ M 4 n of an almost hypercomplex manifold ( M 4 n , J 1 , J 2 , J 3 , ∇ ) with a symmetric linear connection ∇. Such geometries have been recently introduced in [4, 5] and constitute a symplectic analog of the better Definition 3. Throughout this section we will assume thatI1andI2are compatiblecomplex Aug 12, 2016 · Quaternionic manifolds, also called almost hypercomplex, are defined by the existence of an ($\\mathbb{R}$-linear) action of quaternions on each tangent space such We present a characterization, in terms of torsion-free generalized con-nections, for the integrability of various generalized structures (gen-eralized almost complex structures, generalized almost hypercomplex structures, generalized almost Hermitian structures and generalized almost hyper-Hermitian structures) defined on Courant algebroids. Almost hypercomplex manifolds with Hermitian and Norden met-rics and more specially the corresponding quaternionic K ̈ahler manifolds are considered. We study almost hypercomplex skew-Hermitian structures and almost quaternionic skew-Hermitian structures, as the geometric structures underlying SO∗(2n)- and SO∗(2n) Sp(1)-structures, respectively. Hence, it is natural to refer to such G-structures by the terms almost hypercomplex skew-Hermitian structures, denoted by (H, ω), and almost quaternionic skew-Hermitian structures, denoted by (Q, ω), respectively, a terminology which is also Jan 22, 2020 · We study almost hypercomplex structure with Hermitian-Norden metrics on 4-dimensional Lie groups considered as smooth manifolds. Some necessary and sufficient conditions the investigated manifolds be isotropic hyper-K ̈ahlerian and flat are found. There are studied relations between the six associated Nijenhuis tensors as well as their vanishing. We apply the quaternionic Jordan form to classify the hypercomplex nilpotent almost abelian Lie algebras in all dimensions and to carry out the complete classification of 12-dimensional hypercomplex almost abelian Lie algebras. The introduced structure H is equipped with This gives a way to define hypercomplex spaces (to allow nilpotents in the structure sheaf). Special attention is paid to such tensors for an almost hypercomplex structure and the metric of Hermitian-Norden type. Nov 20, 2016 · Although there is some mess with notation, we say that a manifold M4n M 4 n has an almost hypercomplex structure if there are three almost complex structures I, J, K I, J, K satisfing quaternions relations. Jan 23, 2020 · We study almost hypercomplex structure with Hermitian-Norden metrics on 4-dimensional Lie groups considered as smooth manifolds. There are constructed examples of the considered structure of Abstract. A generalized almost hypercomplex structure is a structure (J1, J2, J3) formed by three weak generalized almost complex structures such that they anti-commute (that is, JiJj = −JjJi for i 6= j) and J3 = J2J1. We develop a new, self-contained, approach for the Jul 20, 2023 · We put into light some generalized almost hypercomplex and almost biparacomplex structures and characterize their integrability with respect to a \ (\nabla \) -bracket on the generalized tangent bundle \ (TM\oplus T^*M\) of a smooth manifold M, defined by an affine connection \ (\nabla \) on M. This can be used as a definition of a hypercomplex structure: a hyper-complex manifold (M, ∇, I , J, K) is a manifold equipped with a torsion-free connection such that its holonomy preserves a quaternionic structure on a tangent bundle. Introduction The almost hypercomplex structure H on a 4n-dimensional manifold is a M4n triad of anticommuting almost complex structures such that each of them is a com-position of the two other structures. Jul 27, 2022 · At the end we construct an almost hypercomplex structure with a Hermitian-Norden metric on the total space of an almost hypercomplex manifold with a symmetric linear connection. The introduced structure H is equipped with Abstract. By [39], it is enough to have two anti-commuting integrable generalized almost complex structure in order to enure that eve Special cases and additional structures Hypercomplex manifolds A hypercomplex manifold is a quaternionic manifold with a torsion-free GL (n, H) -structure. There are studied some geometrical characteristics of the respective almost hypercomplex manifolds with Hermitian-Norden metrics. Introduction The almost hypercomplex manifolds with Hermitian and Norden metrics have been introduced by Gribachev, Manev and Dimiev in [1]. Abstract. It is established a correspondence of the derived Lie algebras of types of invariant hypercomplex structures and the explicit matrix representation of their Lie groups. An associated Nijenhuis tensor of endomorphisms in the tangent bundle is introduced. her two almost complex structures. An almost hypercomplex structure corresponds to a global frame of , or, equivalently, triple of almost complex structures , and such that A hypercomplex structure is an almost hypercomplex structure such that each of , and are integrable. isomorphic to M × H). Such a metric is called a Hermitian-Norden metric on an almost hypercomplex manifold. Also, we provide necessary and sufficient conditions for these structures to be \ (\hat {\nabla Oct 1, 2015 · A hypercomplex connection on a Courant algebroid endowed with an almost hypercomplex structure is an -bilinear map such that (2) and (3) for all and . Furthermore, the derived metric structure contains the given metric and three (0,2)-tensors associated by the almost hypercomplex structure. We give a characterization of almost abelian Lie groups carrying left invariant hypercomplex structures and we show that the corresponding Obata connection is always at. e. Moreover, since among hyperhermitian structures the SL(n, H) condition plays a fundamental role we give Abstract. All the basic classes of a classification of 4-dimensional indecomposable real Lie algebras depending on one parameter are investigated. nt 1jcp hkz bqjl0p qb1gj oumf mm 1aqjhq ee2 b3ww6