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

We continue the study of the variety XM] of planar normal sections on a natural embedding of a flag manifold M. Here we consider those subvarieties of XM] 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 TT] (M), invariant by the torus action and which give rise to real projective spaces in XM]. The other one is the following. Let be the tangent space of the inner symmetric space G/K at K] . Then RP ( ) is maximal in XM] if and only if 2(G/K) does not vanish.