**Journal of Lie Theory
**

Vol. 8, No. 2, pp. 415-428 (1998)

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

##
W. Dal Lago, A. Garcia, and C. U. Sanchez

Fa.M.A.F.

Universidad Nacional de Cordoba

Ciudad Universitaria

5000, Cordoba, Argentina.

csanchez@famaf.unc.edu.ar

**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.

**Full text of the article:**

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