Differential geometry of $\mathsf{SO}^\ast(2n)$-type structures

Abstract

We study $4n$-dimensional smooth manifolds admitting a $\mathsf{SO}^*(2n)$- or a $\mathsf{SO}^*(2n)\mathsf{Sp}(1)$-structure, where $\mathsf{SO}^*(2n)$ is the quaternionic real form of $\mathsf{SO}(2n, \mathbb{C})$. We show that such $G$-structures, called almost hypercomplex/quaternionic skew-Hermitian structures, form the symplectic analogue of the better known almost hypercomplex/quaternionic-Hermitian structures (hH/qH for short). We present several equivalent definitions of $\mathsf{SO}^*(2n)$- and $\mathsf{SO}^*(2n)\mathsf{Sp}(1)$-structures in terms of almost symplectic forms compatible with an almost hypercomplex/quaternionic structure, a quaternionic skew-Hermitian form, or a symmetric 4-tensor, the latter establishing the counterpart of the fundamental 4-form in almost hH/qH geometries. The intrinsic torsion of such structures is presented in terms of Salamon's $\mathsf{E}\mathsf{H}$-formalism, and the algebraic types of the corresponding geometries are classified. We construct explicit adapted connections to our $G$-structures and specify certain normalization conditions, under which these connections become minimal. Finally, we present the classification of symmetric spaces $K/L$ with $K$ semisimple admitting an invariant torsion-free $\mathsf{SO}^*(2n)\mathsf{Sp}(1)$-structure. This paper is the first in a series aiming at the description of the differential geometry of $\mathsf{SO}^*(2n)$- and $\mathsf{SO}^*(2n)\mathsf{Sp}(1)$-structures.