Abstract
The space Π of centered m-planes is considered in projective space P n . A principal bundle is associated with the space Π and a group connection is given on the principal bundle. The connection is not uniquely induced at the normalization of the space Π . Semi-normalized spaces Π 1 , Π 2 and normalized space Π 1 , 2 are investigated. By virtue of the Cartan–Laptev method, the dynamics of changes of corresponding bundles, group connection objects, curvature and torsion of the connections are discovered at a transition from the space Π to the normalized space Π 1 , 2 .
Highlights
This paper refers to the field of differential geometry, or, more precisely, to the theory of differentiable manifolds equipped with various “geometric structures” [1], such as connection, curvature and torsion
Upon use of this method, a research of geometry of a manifold with geometric structures fixed on it is reduced to study of geometry of other manifolds
The Cartan–Laptev method is applied to research of the centered m-planes space in projective space Pn
Summary
This paper refers to the field of differential geometry, or, more precisely, to the theory of differentiable manifolds equipped with various “geometric structures” [1], such as connection, curvature and torsion. Universality and efficiency of the Cartan method were shown in many papers Upon use of this method, a research of geometry of a manifold with geometric structures fixed on it is reduced to study of geometry of other manifolds (total space of frames above the given manifold or subbundles of the bundle). The Cartan–Laptev method is applied to research of the centered m-planes space in projective space Pn. The connection theory (see, e.g., [5,6]) has an important place in differential geometry. Lumiste [10] has entered a similar normalization of a manifold of m-planes in projective space An analogue of this normalization is used in this paper. It is important to emphasize that the Grassmann manifold plays a key role in topology and geometry as the base space of an universal vector bundle.
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