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; this is exemplified by the classification of the totally geodesic submanifolds in the complex quadric Q m : = SO ( m + 2 ) / ( SO ( 2 ) × SO ( m ) ) obtained in the second part of the article. The classification shows that the earlier classification of totally geodesic submanifolds of Q m by Chen and Nagano (see [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745–755]) is incomplete. More specifically, two types of totally geodesic submanifolds of Q m are missing from [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745–755]: The first type is constituted by manifolds isometric to C P 1 × R P 1 ; their existence follows from the fact that Q 2 is (via the Segre embedding) holomorphically isometric to C P 1 × C P 1 . The second type consists of 2-spheres of radius 1 2 10 which are neither complex nor totally real in Q m .