Invariant Hypercomplex Structure Research Articles

Hypercomplex Almost Abelian Solvmanifolds

We give a characterization of almost abelian Lie groups carrying left invariant hypercomplex structures and we show that the corresponding Obata connection is always flat. We determine when such Lie groups admit HKT metrics and study the corresponding Bismut connection. We obtain the classification of hypercomplex almost abelian Lie groups in dimension 8 and determine which ones admit lattices. We show that the corresponding 8-dimensional solvmanifolds are nilmanifolds or admit a flat hyper-Kähler metric. Furthermore, we prove that any 8-dimensional compact flat hyper-Kähler manifold is a solvmanifold equipped with an invariant hyper-Kähler structure. We also construct almost abelian hypercomplex nilmanifolds and solvmanifolds in higher dimensions.

Invariant hypercomplex structures and algebraic curves

AbstractWe show that ‐invariant hypercomplex structures on (open subsets) of regular semisimple adjoint orbits in correspond to algebraic curves C of genus , equipped with a flat projection of degree k, and an antiholomorphic involution covering the antipodal map on .

Hypercomplex structures with Hermitian–Norden metrics on four-dimensional Lie algebras

Integrable hypercomplex structures with Hermitian and Norden metrics on Lie groups of dimension 4 are considered. The corresponding five types of invariant hypercomplex structures with hyper-Hermitian metric, studied by Barberis, are constructed here. The different cases regarding the signature of the basic pseudo-Riemannian metric are considered.

Homogeneous hyper-complex structures and the Joyceʼs construction

We prove that any invariant hyper-complex structure on a homogeneous space M = G / L where G is a compact Lie group is obtained via the Joyceʼs construction, provided that there exists a hyper-Hermitian naturally reductive invariant metric on M.

ON SOME HYPERCOMPLEX 4-DIMENSIONAL LIE GROUPS OF CONSTANT SCALAR CURVATURE

In this paper we study sectional curvature of invariant hyper-Hermitian metrics on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. We give the Levi–Civita connections and explicit formulas for computing sectional curvatures of these metrics and show that all these spaces have constant scalar curvature. We also show that they are flat or they have only non-negative or non-positive sectional curvature.

Randers metrics of Berwald type on four-dimensional hypercomplex Lie groups

In the present paper we study Randers metrics of Berwald type on simply connected four-dimensional real Lie groups admitting invariant hypercomplex structure. On these spaces, the Randers metrics arising from invariant hyper-Hermitian metrics are considered. Then we give explicit formulae for computing the flag curvature of these metrics. By this study we construct two four-dimensional Berwald spaces, one of them with a non-negative flag curvature and the other one with a non-positive flag curvature.

DEFORMATIONS OF HYPERCOMPLEX STRUCTURES ASSOCIATED TO HEISENBERG GROUPS

Let X be a compact quotient of the product of the real Heisenberg group H 4m+1 of dimension 4m + 1 and the three-dimensional real Euclidean space R 3 . A left-invariant hypercomplex structure on H 4m+1 × R 3 descends onto the compact quotient X. The space X is a hyperholomorphic fibration of 4-tori over a 4m-torus. We calculate the parameter space and obstructions to deformations of this hypercomplex structure on X. Using our calculations, we show that all small deformations generate invariant hypercomplex structures on X but not all of them arise from deformations of the lattice. This is in contrast to the deformations on the 4m-torus.

Complexification and Hypercomplexification of Manifolds with a Linear Connection

We give a simple interpretation of the adapted complex structure of Lempert–Szöke and Guillemin–Stenzel: it is given by a polar decomposition of the complexified manifold. We then give a twistorial construction of an SO(3)-invariant hypercomplex structure on a neighbourhood of X in TTX, where X is a real-analytic manifold equipped with a linear connection. We show that the Nahm equations arise naturally in this context: for a connection with zero curvature and arbitrary torsion, the real sections of the twistor space can be obtained by solving Nahm's equations in the Lie algebra of certain vector fields. Finally, we show that, if we start with a metric connection, then our construction yields an SO(3)-invariant hyperkähler metric.

Hypercomplex eight-dimensional nilpotent Lie groups

In this paper we give a description of all eight-dimensional simply connected nilpotent Lie groups carrying a left invariant hypercomplex structure.

Affine connections on homogeneous hypercomplex manifolds

It is the aim of this work to study affine connections whose holonomy group is contained in Gl(n, H) . These connections arise in the context of hypercomplex geometry. We study the case of homogeneous hypercomplex manifolds and introduce an affine connection which is closely related to the Obata connection [M. Obata, Japan J. Math. 26 (1956) 43–77]. We find a family of homogeneous hypercomplex manifolds whose corresponding connections are not flat with holonomy contained in Sl(n, H) . We consider first the 4-dimensional case and determine all the 4-dimensional real Lie groups which admit integrable invariant hypercomplex structures. We describe explicitly the Obata connection corresponding to these structures and by studying the vanishing of the curvature tensor, we determine which structures are integrable, obtaining as a byproduct a self-dual, non-flat, Ricci flat affine connection on R 4 admitting a simply transitive solvable group of affine transformations. This result extends to a family of hypercomplex manifolds of dimension 4 n, n > 1, considered in [M.L. Barberis, I.D. Miatello, Quart. J. Math. Oxford 47 (2) (1996) 389–404]. We also give a sufficient condition for the integrability of hypercomplex structures on certain solvable Lie algebras.

Hypercomplex structures on four-dimensional Lie groups

The purpose of this paper is to classify invariant hypercomplex structures on a 4 4 -dimensional real Lie group G G . It is shown that the 4 4 -dimensional simply connected Lie groups which admit invariant hypercomplex structures are the additive group H \mathbb H of the quaternions, the multiplicative group H ∗ {\mathbb H}^* of nonzero quaternions, the solvable Lie groups acting simply transitively on the real and complex hyperbolic spaces, R H 4 {\mathbb R}H^4 and C H 2 {\mathbb C}H^2 , respectively, and the semidirect product C ⋊ C {\mathbb C}\rtimes {\mathbb C} . We show that the spaces C H 2 {\mathbb C}H^2 and C ⋊ C {\mathbb C}\rtimes {\mathbb C}\, possess an R P 2 {\mathbb R}P^2 of (inequivalent) invariant hypercomplex structures while the remaining groups have only one, up to equivalence. Finally, the corresponding hyperhermitian 4 4 -manifolds are determined.