Totally geodesic submanifolds of the complex quadric

Abstract

In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Riemannian symmetric spaces. In this way a classification of the totally geodesic submanifolds in the complex quadric $Q^m := \SO(m+2)/(\SO(2) \times \SO(m))$ is obtained. It turns out that the earlier classification of totally geodesic submanifolds of $Q^m$ by Chen and Nagano is incomplete: in particular a type of submanifolds which are isometric to 2-spheres of radius $\frac{1}{2}\sqrt{10}$, and which are neither complex nor totally real in $Q^m$, is missing.

22 pages

Cited by in corpus (7)

• Contact real hypersurfaces in the complex hyperbolic quadric
• Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagram
• Totally geodesic submanifolds of the complex and the quaternionic 2-Grassmannians
• Commuting Jacobi operators on Real hypersurfaces of Type B in the complex quadric
• Constant Angle Surfaces in Product Spaces
• Real hypersurfaces in the complex quadric with commuting Ricci tensor
• Derivatives of normal Jacobi operator on real hypersurfaces in the complex quadric