Publication details
Differential geometry of SO*(2n)-type structures
| Authors | |
|---|---|
| Year of publication | 2022 |
| Type | Article in Periodical |
| Magazine / Source | Annali di Matematica Pura ed Applicata |
| MU Faculty or unit | |
| Citation | |
| web | https://link.springer.com/article/10.1007/s10231-022-01212-y |
| Doi | http://dx.doi.org/10.1007/s10231-022-01212-y |
| Keywords | Quaternionic real form; Almost hypercomplex structures; Almost quaternionic structures; Almost hypercomplex skew-Hermitian structures; Almost quaternionic skewHermitian structures; Skew-Hermitian quaternionic forms; Scalar 2-forms; Intrinsic torsion |
| Description | We study 4n-dimensional smooth manifolds admitting a SO*(2 n) - or a SO*(2 n) Sp(1) -structure, where SO*(2 n) is the quaternionic real form of SO(2 n, 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 SO*(2 n) - and SO*(2 n) 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 EH-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 SO*(2 n) Sp(1) -structure. This paper is the first in a series aiming at the description of the differential geometry of SO*(2 n) - and SO*(2 n) Sp(1) -structures. |