The composition equipment for congruence of hypercentred planes

Abstract

In n-dimensional projective space Pn a manifold , i. e., a cong­ruence of hypercentered planes , is considered. By a hypercentered planе we mean m-dimensional plane with a (m – 1)-dimensional hy­perplane , distinguished in it. The first-order fundamental object of the congruence is a pseudotensor. The principal fiber bundle is associated with the congru­ence, . The base of the bundle is the manifold and a typical fiber is the stationarity subgroup of a centered plane . In principal fiber bundle a fundamental-group connection is given us­ing the field of the object . The composition equipment for the congruence is 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. The composition equipment is given by field of quasitensor . It is proved that the composition equipment for the congruence of hypercentred m-planes induces a fundamental-group connection with object in the principal bundle associated with the con­gruence. In proof, the envelopments are built for the com­ponents of the connection object .

Keywords

• projective space
• hypercentred plane
• congruence
• connec­tion in a principal fiber bundle
• composition equipment