Parallel displacements on the surface of a projective space

Reduction of Bundles, Connection, Curvature, and Torsion of the Centered Planes Space at Normalization

“…Now, suppose that we have already made canonizations considered in items 4.1 and 4.2 simultaneously, that is, A α ∈ N n−m and A a ∈ N m−1 . In this case, conditions (8) and (14) are satisfied and the structure Equation (2) will become…”

“…Without using the previous canonization and placing the vertices A a on the second normal N m−1 , we get condition (14). The connection object Γ is narrowed to the object Γ 2 = Γ \ (Γ 3 \ Γ 1 ) with the following congruences for its components:…”

“…By substituting conditions (14) into the differential congruences for components of the connection object Γ, we get congruences (26) with conditions…”

“…Taking into account conditions (8) and (14) in the first six and last three differential congruences (24), we have that the components of the reduced connection object Γ I,I I satisfy the differential congruences (27).…”

“…Thus, in our case, the normal of the first kind (the first normal) is a subspace N n−m of P n having only one common point with a centered m-plane T m ; and the normal of the second kind (the second normal) is a subspace N m−1 of the centered m-plane T m not passing through its centre [11]. Moreover, we will also use a reduction that is frequently applied in geometry (see, e.g., [4,[12][13][14][15][16][17]). This paper is a continuation of author's research [18,19].…”