The composite equipment for manifold of hypercentered planes, whose dimension coincides with dimension of generating plane

Abstract

In n-dimensional projective space Pn a manifold , i. e., a family of pairs of planes one of which is a hyperplane in the other, is considered. A principal bundle arises over it, . A typi­cal fiber is the stationarity subgroup of the generator of pair of planes: external plane and its multidimensional center — hyperplane. The princi­pal bundle contains four factor-bundles. A fundamental-group connection is set by the Laptev — Lumiste method in the associated fibering. It is shown that the connection object contains four subobjects that define connections in the corresponding fac­tor-bundles. It is proved that the curvature object of fundamental-group connection forms pseudotensor. It contains four subpseudotensors, which are curvature objects of the corresponding subconnections. The composite equipment of the family of hypercentered planes set by means of a point lying in the plane and not belonging to its hypercent­er and an (n – m – 1)-dimensional plane, which does not have common points with the hypercentered plane. It is proved, that composite equip­ment induces the fundamental-group connections of two types in the as­sociated fibering.

Keywords

• projective space
• family of hypercentered planes
• fundamental-group connection
• curvature
• pseudotensor