Centered planes in the projective connection space

• Published in: Differential Geometry of Manifolds of Figures
• Main Author: O.O. Belova
• Format: Article in Journal/Newspaper
• Language: English; Russian
• Published: Immanuel Kant Baltic Federal University 2020
• Subjects: projective connection space; space of centered planes; fibering; connection; Mathematics; QA1-939; laptev
• Online Access: https://doi.org/10.5922/0321-4796-2020-51-4; https://doaj.org/article/047eea12e8234c129b1d8a835e69f592

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.