Hypersurfaces in the noncompact Grassmann manifold SU_{2,m}/S(U_2U_m)

TL;DR

This paper studies hypersurfaces in a complex and quaternionic symmetric space, focusing on how their tangent bundle substructures relate to their shape, advancing understanding of geometric properties in such spaces.

Contribution

It introduces new insights into the geometric structure of hypersurfaces in SU_{2,m}/S(U_2U_m), especially regarding the interaction of complex and quaternionic subbundles with the hypersurface shape.

Findings

Characterization of hypersurfaces with specific subbundle properties

Relations between complex and quaternionic structures and hypersurface curvature

New classifications of hypersurfaces based on subbundle interactions

Abstract

The Riemannian symmetric space SU_{2,m}/S(U_2U_m) is both Hermitian symmetric and quaternionic Kahler symmetric. Let M be a hypersurface in SU_{2,m}/S(U_2U_m) and denote by TM its tangent bundle. The complex structure of SU_{2,m}/S(U_2U_m) determines a maximal complex subbundle C of TM, and the quaternionic structure of SU_{2,m}/S(U_2U_m) determines a maximal quaternionic subbundle Q of TM. In this article we investigate hypersurfaces in SU_{2,m}/S(U_2U_m) for which C and Q are closely related to the shape of M.