Differential geometry of ${\mathsf{SO}}^{*}(2n)$-structures and ${\mathsf{SO}}^{*}(2n){\mathsf{Sp}}(1)$-structures - Part II

Ioannis Chrysikos, Jan Gregorovič, Henrik Winther

Published 2021-11-08Version 1

In the first part of this series of articles we studied almost hypercomplex skew-Hermitian structures and almost quaternionic skew-Hermitian structures, as those underlying geometric structures on $4n$-dimensional oriented manifolds appearing as $G$-structures for the Lie groups ${\mathsf{SO}}^*(2n)$ and ${\mathsf{SO}}^{*}(2n){\mathsf{Sp}}(1)$, respectively. There the corresponding intrinsic torsion is computed and the number of algebraic types of related geometries is derived, together with the minimal adapted connections (with respect to certain normalizations conditions). Here we use these results to present the related first-order integrability conditions in terms of the several algebraic types and other constructions. In particular, we use distinguished connections to provide a more geometric interpretation of the presented integrability conditions and highlight some features of certain classes. The second main contribution of this note is the illustration of several specific types of such geometries via a variety of examples. We present an analogue of the notion of quaternionification of vector spaces, at the level of manifolds, which use to describe two general constructions providing examples of ${\mathsf{SO}}^*(2n)$-structures. We also use the bundle of Weyl structures and describe examples of ${\mathsf{SO}}^{*}(2n){\mathsf{Sp}}(1)$-structures in terms of parabolic geometries.