Planar normal sections on the natural imbedding of a flag manifold

We study the setX[M] of planar normal sections on the natural imbedding of a flag manifoldM. We characterizeX[M] and show that it is a real algebraic submanifold of Pⁿ (n=dimM). From results of Chen and Ferus, it may be concluded thatX[M] measures how farM is from a symmetricR-space. We compute the Euler characteristic ofX[M] and its complexificationX_c[M]. Our main result shows, in particular, thatx(X[M]) andx(X_c[M]) depend only on dimM and not on the nature ofM itself.