Centered planes in the projective connection space

Abstract

The space of centered planes is considered in the Cartan projec­ti­ve connection space . The space is important because it has con­nec­tion with the Grassmann manifold, which plays an important role in geometry and topology, since it is the basic space of a universal vector bundle. The space is an n-dimensional differentiable manifold with each point of which an n-dimensional projective space containing this point is associated. Thus, the manifold is the base, and the space is the n-dimensional fiber “glued” to the points of the base. A projective space is a quotient space of a linear space with respect to the equivalence (collinearity) of non-zero vectors, that is . The projective space is a manifold of di­men­sion n. In this paper we use the Laptev — Lumiste invariant analytical meth­od of differential geometric studies of the space of centered planes and introduce a fundamental-group connection in the associated bundle . The bundle contains four quotient bundles. It is show that the connection object is a quasi-tensor containing four subquasi-tensors that define connections in the corresponding quotient bundles.

Keywords

• projective connection space
• space of centered planes
• fibering
• connection