Hypercomplex structures on Stiefel manifolds

Abstract

This paper describes a family of hypercomplex structures {ZO(p)}a=,2,3 depending on n real non-zero parameters p = (Pl,... ,n) on the Stiefel manifold of complex 2-planes in C" for all n > 2. Generally, these hypercomplex structures are inhomogeneous with the exception of the case when all the pi's are equal. We also determine the Lie algebra of infinitesimal hypercomplex automorphisms for each structure. Furthermore, we solve the equivalence problem for the hypercomplex structures in the case that the components of p are pairwise commensurable. Finally, some of these examples admit discrete hypercomplex quotients whose topology we also analyze.

Geometry of Hyper-Kähler Connections with Torsion

“…Since the metric is hyper-Hermitian, for any vectors X and Y , F 1 (X, Y ) = F 2 (I 3 X, Y ). Through the integrability of the complex structures I 1 , I 2 , I 3 , the quaternion identities (5) and the last two identities in (21), one derives the first identity in (21). Therefore, we have the following theorem which justifies our definition for potential functions.…”

“…There are homogeneous examples [18]. There are inhomogeneous examples [5] [22]. There is a reduction construction modeled on symplectic reduction and hyper-Kähler reduction [17].…”

Some Constructions of Almost Para-Hyperhermitian Structures on Manifolds and Tangent Bundles

“…An almost para-hermitian structure on a differentiable manifold M is a pair (P, g), where P is an almost product structure on M and g is a semi-Riemannian metric on M satisfying: (8) g(P X, P Y ) = −g(X, Y ), for all vector fields X,Y on M . In this case, (M, P, g) is said to be an almost para-hermitian manifold.…”

Cohomologies on Hypercomplex Manifolds

“…Furthermore, Dominic Joyce constructed many left-invariant hypercomplex structures on Lie groups [16] and similar ones have been analysed by physicists interested in string theory [30] in the context of N = 4 supersymmetry. In more recent years, various authors constructed inhomogeneous hypercomplex structures: see for example [9] for hypercomplex structures on Stiefel manifolds as well as [7,27].…”