About differential equations of the curvature tensors of a fundamental group and affine connections

2019 ◽  
pp. 133-140
Author(s):  
N. Ryazanov
Keyword(s):  
Differential Equations ◽  
Fundamental Group ◽  
Curvature Tensor ◽  
Principal Bundle ◽  
Group Structure ◽  
Affine Connection ◽  
Affine Connections ◽  
First Order ◽  
Structure Equations ◽  
Curvature Tensors

The principal bundle is considered, the base of which is an n-dimensional smooth manifold, and the typical fiber is an r-fold Lie group. Structure equations for the forms of the fundamental group and affine connections are given, each of which contains the corresponding components of the curvature tensor. For each connection, an approach is shown that allows to find the differential equations for the components of the curvature tensor of the corresponding connection in a faster way than by differentiating the expressions of these objects in terms of the connection objects and their Pfaffian derivatives. The method consists in successively solving cubic equations, first by Laptev’s lemma, then by Cartan’s lemma. Taking into account the comparisons modulo basic forms, we obtain already known results (see [3]). Thus, differential equations are derived for the components of the curvature tensor of the first-order fundamentalgroup connection, as well as for the components of the curvature tensor of the affine connection.

Curvature-torsion quasitensor of Laptev fundamental-group connection

2020 ◽  
pp. 155-169
Author(s):  
Yu. I. Shevchenko
Keyword(s):  
Fundamental Group ◽  
Curvature Tensor ◽  
Principal Bundle ◽  
Torsion Tensor ◽  
Structural Equations ◽  
The Other ◽  
Cartan Connection ◽  
Cartan Connections ◽  
Special Cases ◽  
The Given

We consider a space with Laptev's fundamental group connection generalizing spaces with Cartan connections. Laptev structural equations are reduced to a simpler form. The continuation of the given structural equations made it possible to find differential comparisons for the coeffi­cients in these equations. It is proved that one part of these coefficients forms a tensor, and the other part forms is quasitensor, which justifies the name quasitensor of torsion-curvature for the entire set. From differential congruences for the components of this quasitensor, congruences are ob­tained for the components of the Laptev curvature-torsion tensor, which contains 9 subtensors included in the unreduced structural equations. In two special cases, a space with a fundamental connection is a spa­ce with a Cartan connection, having a quasitensor of torsion-curvature, which contains a quasitensor of torsion. In the reductive case, the space of the Cartan connection is turned into such a principal bundle with connec­tion that has not only a curvature tensor, but also a torsion tensor.

scholarly journals Generalized Affine Connections Associated with the Space of Centered Planes

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 782
Author(s):  
Olga Belova
Keyword(s):  
Projective Space ◽  
Affine Connection ◽  
Affine Connections ◽  
Curvature Tensors

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.

scholarly journals Some properties of ET-projective tensors obtained from Weyl projective tensor

Filomat ◽  
2015 ◽  
Vol 29 (3) ◽  
pp. 573-584
Author(s):  
Mica Stankovic ◽  
Milan Zlatanovic ◽  
Nenad Vesic
Keyword(s):  
Curvature Tensor ◽  
Affine Connection ◽  
Connection Coefficients ◽  
Linearly Independent ◽  
Projective Curvature ◽  
Projective Tensor ◽  
Connection Space ◽  
Curvature Tensors

Vanishing of linearly independent curvature tensors of a non-symmetric affine connection space as functions of vanished curvature tensor of the associated space of this one are analyzed in the first part of this paper. Projective curvature tensors of a non-symmetric affine connection space are expressed as functions of the affine connection coefficients and Weyl projective tensor of the corresponding associated affine connection space in the second part of this paper.

Curvature tensor of connection in principal bundle of Cartan's projective connection space

2019 ◽  
pp. 36-40 ◽  
Author(s):  
K. Bashashina
Keyword(s):  
Projective Space ◽  
Curvature Tensor ◽  
Principal Bundle ◽  
Torsion Tensor ◽  
Projective Connection ◽  
Affine Curvature ◽  
Structure Equations ◽  
Connection Space

We considered Cartan's projective connection space with structure equations generalizing the structure equations of the projective space and the condition of local projectivity (this condition is an analogue to the equiprojectivity condition in the projective space). The curvature-torsion object of the space is a tensor containing three subtensor: torsion tensor, torsion affine curvature tensor, extended torsion tensor. Cartan's projective connection space is not a space with connection of the principal bundle. The assignment of a connection in the adjoint principal bundle leads to a space with a connection. It is proved that the curvature object of the introduced connection is a tensor.

The composition equipment for congruence of hypercentred planes

2019 ◽  
pp. 61-67
Author(s):  
A. V. Vyalova
Keyword(s):  
Projective Space ◽  
Fundamental Group ◽  
Fiber Bundle ◽  
Principal Bundle ◽  
First Order ◽  
Principal Fiber Bundle ◽  
Dimensional Plane ◽  
Fundamental Object ◽  
Dimensional Projective Space

In n-dimensional projective space Pn a manifold Vnm , i. e., a congruence of hypercentered planes Pm , is considered. By a hypercentered planе Pm we mean m-dimensional plane with a (m – 1)-dimensional hyperplane Lm1 , distinguished in it. The first-order fundamental object  of the congruence is a pseudotensor. The principal fiber bundle Gr (Vnm) is associated with the congruence, r  n(n m1)  m2. . The base of the bundle is the manifold Vnm and a typical fiber is the stationarity subgroup Gr of a centered plane Pm . In principal fiber bundle a fundamental-group connection is given using the field of the object Г . The composition equipment for the congruence is set by means of a point lying in the plane and not belonging to its hypercenter and an (n – m – 1)-dimensional plane, which does not have common points with the hypercentered plane. The composition equipment is given by field of quasitensor  . It is proved that the composition equipment for the congruence Vnm of hypercentred m-planes Pm induces a fundamental-group connection with object Г in the principal bundle Gr (Vnm ) associated with the congruence. In proof, the envelopments Г  Г(, ) are built for the components of the connection object Г .

Canonical Almost Geodesic Mappings of the First Type of Spaces with Affine Connection Onto Generalized 2-Ricci-Symmetric Spaces

2021 ◽  
Vol 22 ◽  
pp. 78-87
Author(s):  
Volodymyr Berezovski ◽  
Yevhen Cherevko ◽  
Svitlana Leshchenko ◽  
Josef Mikes
Keyword(s):  
Differential Equations ◽  
Closed System ◽  
Symmetric Spaces ◽  
Affine Connection ◽  
Linear Differential Equations ◽  
Cauchy Type ◽  
Affine Connections ◽  
Covariant Derivatives ◽  
Linear Differential

In the paper we consider almost geodesic mappings of the first type of spaces with affine connections onto generalized 2-Ricci-symmetric spaces. The main equations for the mappings are obtained as a closed system of linear differential equations of Cauchy type in the covariant derivatives. The obtained result extends an amount of research produced by Sinyukov, Berezovski and Mike\v{s}.

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