Maximal Projective Subspaces in the Variety of Planar Normal Sections of a Flag Manifold

Abstract

We continue the study of the variety X[M] of planar normal sections on a natural embedding of a flag manifold M. Here we consider those subvarieties of X[M] that are projective spaces. When M=G/T is the manifold of complete flags of a compact simple Lie group G, we obtain our main results. The first one characterizes those subspaces of the tangent space T[T] (M), invariant by the torus action and which give rise to real projective spaces in X[M]. The other one is the following. Let $\mathfrak{p}$ be the tangent space of the inner symmetric space G/K at [K] . Then RP ($\mathfrak{p}$) is maximal in X[M] if and only if π2(G/K) does not vanish.