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Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations -
J. Slovák, A. Čap (2009)
Parabolic Geometries I
- Publisher
- Springer Journals
- Copyright
- Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022
- ISSN
- 1664-2368
- eISSN
- 1664-235X
- DOI
- 10.1007/s13324-022-00701-w
- Publisher site
- See Article on Publisher Site
Abstract
We study almost hypercomplex skew-Hermitian structures and almost quaternionic skew-Hermitian structures, as the geometric structures underlying SO∗(2n)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathsf {SO}}^*(2n)$$\end{document}- and SO∗(2n)Sp(1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathsf {SO}}^*(2n)\mathsf {Sp} (1)$$\end{document}-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 SO∗(2n)Sp(1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathsf {SO}}^*(2n)\mathsf {Sp} (1)$$\end{document}-structures in terms of functorial constructions in the context of parabolic geometries.
Journal
Analysis and Mathematical Physics – Springer Journals
Published: Aug 1, 2022
Keywords: Almost hypercomplex/quaternionic structures; Almost hypercomplex/quaternionic skew-Hermitian structures; Adapted connections; Torsion types; Integrability conditions; Bundle of Weyl structures; 53C10; 53C26; 53D15; 53B05; 53C30; 53A55