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 2spheres of radius $\tfrac{1}{2}\sqrt{10}$, and which are neither complex nor totally real in $Q^m$, is missing.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 2006
 arXiv:
 arXiv:math/0603167
 Bibcode:
 2006math......3167K
 Keywords:

 Mathematics  Differential Geometry;
 53C35 (Primary);
 53C17
 EPrint:
 22 pages