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 $\tfrac{1}{2}\sqrt{10}$, and which are neither complex nor totally real in $Q^m$, is missing.Comment: 22 page

Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagram

“…I have implemented the algorithms and equations given here as a Maple package, which can be found at http:// satake.sourceforge.net. In a forthcoming article, I will use this presentation to classify the totally geodesic submanifolds in the exceptional Riemannian symmetric spaces of rank 2 (based on similar methods as my classification in the 2-Grassmannians, see [4] and [5]). …”

Real hypersurfaces in the complex quadric with commuting and parallel Ricci tensor

“…For more background to this section we refer to [2], [5], [10], [16], and [18]. The complex quadric Q m is the complex hypersurface in CP m+1 which is defined by the equation z with constant curvature, and Q 2 is isometric to the Riemannian product of two 2-spheres with constant curvature.…”

Real hypersurfaces in the complex quadric with commuting Ricci tensor

“…For more background to this section we refer to [2], [5], [10], [16], and [18]. The complex quadric Q m is the complex hypersurface in CP m+1 which is defined by the equation z 2 0 + · · ·+ z 2 m+1 = 0, where z 0 , .…”

“…Accordingly, the complex quadric admits two important geometric structures, a complex conjugation structure A and a Kähler structure J, which anti-commute with each other, that is, AJ = −JA. Then for m≥2 the triple (Q m , J, g) is a Hermitian symmetric space of compact type with rank 2 and its maximal sectional curvature is equal to 4 (see Klein [2] and Reckziegel [10]). …”