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

TL;DR

This paper investigates the integrability conditions of special geometric structures related to (2n) and (2n)(1), providing algebraic and geometric characterizations along with illustrative examples.

Contribution

It derives explicit first-order integrability conditions for (2n) and (2n)(1)-structures using intrinsic torsions and distinguished connections, and offers concrete examples within parabolic geometries.

Findings

Derived algebraic types of geometries and minimal adapted connections.

Presented integrability conditions in algebraic and geometric terms.

Constructed explicit examples of (2n)(1)-structures using functorial methods.

Abstract

We study almost hypercomplex skew-Hermitian structures and almost quaternionic skew-Hermitian structures, as the geometric structures underlying $\mathsf{SO}^\ast(2n)$- and $\mathsf{SO}^\ast(2n)\mathsf{Sp}(1)$-structures, respectively. The corresponding intrinsic torsions were computed in the previous article in this series, and the algebraic types of the geometries were 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 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 use the bundle of Weyl structures and describe examples of $\mathsf{SO}^\ast(2n)\mathsf{Sp}(1)$-structures in terms of functorial constructions in the context of parabolic geometries.