Flat connections in a domain of projective space
Abstract
The theory of connections of manifolds has wide application in mathematics and physics.
Flat connection affects the type of parallel translations.
In the present paper, we study the curvature and torsion of connections on the Grassmann manifold of points.
In projective space, the region described by a point is considered. A principal bundle arises over the domain, the typical fiber of which is the stationarity subgroup of a point.
A fundamental group connection according to G. F. Laptev is given in this bundle. It is shown that the Bortolotti’s clothing of the domain induces centroprojective connections of three types in the associated bundle, and these connections have zero curvature and the connections are torsion-free.
A geometric characteristics of the resulting connections are given:
1) when the covariant differentials vanish in all three connections, the clothing hyperplane is immovable;
2) a simple subobject of connection objects , , is characterized by the central projection of the plane adjacent to the hyperplane to the original plane from the center — the point .
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