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• W3204607675 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 understood almost hypercomplex/quaternionic-Hermitian structures (hH/qH in short), where the underlying pseudo-Riemannian geometry is replaced by an almost symplectic geometry. 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 homogeneous 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." @default.
• W3204607675 created "2021-10-11" @default.
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• W3204607675 date "2021-11-08" @default.
• W3204607675 modified "2023-09-27" @default.
• W3204607675 title "Differential geometry of ${mathsf{SO}}^{*}(2n)$-structures and ${mathsf{SO}}^{*}(2n){mathsf{Sp}}(1)$-structures - Part II" @default.
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