Projective subspaces in the variety of normal sections and tangent spaces to a symmetric space

Abstract: In the present article we continue the study of the variety $X\left[ M\right] $ associated to pointwise planar normal sections of a natural imbedding for a flag manifold $M$.

When $M=G/T$ is the manifold of complete flags of a compact simple Lie group $G,$ we obtain two results about subspaces of the tangent space $T_{\left[ T\right] }\left( M\right) $, invariant by the torus action, which give rise to real projective spaces in $X\left[ M\right] $. The first result determines their maximal dimension. While the other one characterizes those of maximal dimension as tangent spaces to the inner symmetric space $G/K$ (the one of largest dimension for the group $G$) at a fixed point of the natural action of the torus $T.$

The last section contains a nice application of these results.

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