Totally geodesic submanifolds of the complex quadric

Details Totally geodesic submanifolds of the complex quadric Totally geodesic submanifolds of the complex quadric

2006-03-07

arxiv.org/abs/math/0603167v1

Differential Geom. Appl. 26 (2008), 79-96

Mathematics

Differential Geometry

22 pages

Scientific paper

10.1016/j.difgeo.2007.11.004

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

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