GEOMETRY OF DISTRIBUTION OF FLAGS IN THE EVEN-DIMENSIONAL AFFINE SPACE

Author:

Kazimieras~NAVICKIS

2010 Mathematics Subject Classification:

91G10.

Keywords:

distribution of m-planes,  Grassmann manifold, normalization of distribution.

Abstract:

The set Gr(m,2m+2)  of all m-planes of the  (2m+2)-dimensional affine space A_{2m+2} is a differentiable manifold. The dimension of this manifold equals  \Gamma=(m+1)(m+2). The manifold Gr(m,2m+2) is called a Grassmann manifold. A generalization of Grassmann manifold is a flag manifold (see [1]--[4]). We call a flag of type (m,m+1)  in the space A_{2m+2}  any pair l_m\subset \Pi_{m+1} of subspaces of this space (\dim l_m=m, \dim \Pi_{m+1}=m+1).

In this note, we construct the differential geometry of distribution of flags of type (m, m+1) on the Grassmann manifold Gr(m,2m+2) in an invariant analytic form. The note is a development of the author's paper [5]. Some intrinsic normalizations of distribution of flags in the affine space are considered.

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Navi-2012

Vol. 7 (15), 2011