Planar Normal Sections on the Natural Embedding of a Real Flag Manifold

Abstract: \font\frak=eufm10 This paper contains a proof of the following result which is an extension of the main result in [DGS], to the general case of real flag manifolds (also called R-spaces or orbits of s-representations).

THEOREM. {\it Let $M$ be a real flag manifold and let $j:M\rightarrow\hbox{\frak p}$ its canonical embedding. Let $X\left[ M\right] \subset RP^{n-1}, (n=\dim M)$ be the variety of directions of pointwise planar normal sections at a point of $M$ and let $X_{c}\left[ M\right] \subset CP^{n-1}$ be the natural complexification of $X\left[ M\right]$. Let $\chi$ denote the Euler-Poincare characteristic then} \smallskip \quad (i)\quad $ \chi(X[M])  =\chi(RP^{n-1})= \cases{0&$n$ even\cr 1&$n$ odd\cr}$

\quad (ii)\quad $\chi(X_c[M])=\chi(CP^{n-1})=n$. \medskip\item{[DGS]} Dal Lago, W.; Garcia, A.; Sanchez, C.U.: Planar Normal Sections on the Natural Imbedding of a Flag Manifold. Geom. Dedicata 53 (1994), 223-235.

Classification (MSC2000): 53C30, 53C42

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