Submanifolds in the variety of planar normal sections

Walter N. Dal Lago, Alicia N. García and Cristián U. Sánchez

Abstract: The present paper continues the study of the nature of the variety $X\left[M\right]$ of directions of pointwise planar normal sections for the manifold $M$ of complete flags of a compact simple Lie group $G_{u}$. The main results concern submanifolds embedded in $RP^{m-1}$ ($m=\dim M$) which are subsets of $X\left[ M\right]$. One of them is an open set in the natural topology of $X\left[M\right]$ whose dimension is related to that of $M$ and the rank of the Lie group $G_{u}$. Others are projective subspaces of ``minimal'' dimension contained in $X\left[M\right]$ for the groups $G_{u}=SU(n+1)$.

Classification (MSC2000): 53C30, 53C42, 32V30

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