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

• Published in: Differential Geometry of Manifolds of Figures
• Main Authors: A.V. Vyalova, Yu. I. Shevchenko
• Format: Article in Journal/Newspaper
• Language: English; Russian
• Published: Immanuel Kant Baltic Federal University 2021
• Subjects: projective space; family of hypercentered planes; fundamental-group connection; curvature; pseudotensor; Mathematics; QA1-939; laptev
• Online Access: https://doi.org/10.5922/0321-4796-2021-52-6; https://doaj.org/article/200119ab64fa4d3881139286180a3648

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.