The space of centered planes and generalized bilinear connection
Abstract
We continue to study the space of centered planes in $n$-dimension projective space. We use E.~Cartan's method of external forms and the group-theoretical method of G.\,F.~Laptev to study the space of centered planes of the same dimension. These methods are successfully applied in physics.
In a generalized bundle, a bilinear connection associated with a space is given. The connection object contains two simplest subtensors and subquasi-tensors (four simplest and three simple subquasi-tensors).
The object field of this connection defines the objects of torsion $S$, curvature-torsion $T$, and curvature $R$. The curvature tensor contains six simplest and four simple subtensors, and curvature-torsion tensor contains three simplest and two simple subtensors.
The canonical case of a generalized bilinear connection is considered.
We realize the strong Lumiste’s affine clothing (it is an analog of the strong Norden’s normalization of the space of centered planes). Covariant differentials and covariant derivatives of the clothing quasi-tensor are described. The covariant derivatives do not form a tensor. We present a geometrical characterization of the generalized bilinear connection using mappings.
In a generalized bundle, a bilinear connection associated with a space is given. The connection object contains two simplest subtensors and subquasi-tensors (four simplest and three simple subquasi-tensors).
The object field of this connection defines the objects of torsion $S$, curvature-torsion $T$, and curvature $R$. The curvature tensor contains six simplest and four simple subtensors, and curvature-torsion tensor contains three simplest and two simple subtensors.
The canonical case of a generalized bilinear connection is considered.
We realize the strong Lumiste’s affine clothing (it is an analog of the strong Norden’s normalization of the space of centered planes). Covariant differentials and covariant derivatives of the clothing quasi-tensor are described. The covariant derivatives do not form a tensor. We present a geometrical characterization of the generalized bilinear connection using mappings.
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