Generalized Affine Connections Associated with the Space of Centered Planes

Belova, Olga

doi:10.3390/math9070782

Mathematics

2021

DOI: 10.3390/math9070782

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Generalized Affine Connections Associated with the Space of Centered Planes

Olga Belova

Abstract: Our purpose is to study a space Π of centered m-planes in n-projective space. Generalized fiberings (with semi-gluing) are investigated. Planar and normal affine connections associated with the space Π are set in the generalized fiberings. Fields of these affine connection objects define torsion and curvature tensors. The canonical cases of planar and normal generalized affine connections are considered.

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Cited by 4 publications

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The space of centered planes and generalized bilinear connection

Belova

2023

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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 Norde?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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The space of centered planes and generalized bilinear connection

Belova

2023

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About the torsion tensor of an affine connection on two-dimensional and three-dimensional manifolds

Polyakova

2021

Differ. Geom. Mnogoobr. Figur

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The basis for this study of affine connections in linear frame bundle over a smooth manifold is the structure equations of the bundle. An affine connection is given in this bundle by the Laptev — Lumiste method. The differential equations are written for components of the deformation tensor from an affine connection to the symmetrical canonical one. The expressions for the components of the torsion tensor for two-dimensional and three-dimensional manifolds were found. For a two-dimensional manifold, the affine torsion is a fraction, in the numerator there is a linear combination of two fiber coordinates which coefficients are two functions depending on the base coordinates (the coordinates on the base), and in the denominator there is the determinant composed of the fiber coordinates (the coordinates in a fiber). For a three-dimensional manifold, the arbitrariness of the numerator is determined by nine functions depending on the base coordinates.

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Generalized bilinear connection on the space of centered planes

Belova

2022

Differ. Geom. Mnogoobr. Figur

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We continue to study the space of centered planes in projective space . In this paper, 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, curvature-torsion, and curvature. 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.

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Generalized Affine Connections Associated with the Space of Centered Planes

Cited by 4 publications

References 28 publications

The space of centered planes and generalized bilinear connection

The space of centered planes and generalized bilinear connection

About the torsion tensor of an affine connection on two-dimensional and three-dimensional manifolds

Generalized bilinear connection on the space of centered planes

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