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.

Сurvature-torsion tensor for Cartan connection

2019 ◽  
pp. 155-168
Author(s):  
Yu. Shevchenko
Keyword(s):  
Homogeneous Space ◽  
Lie Group ◽  
Principal Bundle ◽  
Torsion Tensor ◽  
Projective Connection ◽  
Cartan Connection ◽  
Bianchi Identities ◽  
Covariant Derivatives ◽  
Structure Equations ◽  
Derivatives Of

A Lie group containing a subgroup is considered. Such a group is a principal bundle, a typical fiber of this principal bundle is the subgroup and a base is a homogeneous space, which is obtained by factoring the group by the subgroup. Starting from this group, we constructed structure equations of a space with Cartan connection, which generalizes the Cartan point projective connection, Akivis’s linear projective connection, and a plane projective connection. Structure equations of this Cartan connection, containing the components of the curvature-torsion object, allowed: 1) to show that the curvature-torsion object forms a tensor containing a torsion tensor; 2) to find an analogue of the Bianchi identities such that the curvature-torsion tensor and its Pfaff derivatives satisfy this analogue; 3) to obtain the conditions for the transformation of Pfaffian derivatives of the curvature-torsion tensor into covariant derivatives with respect to the Cartan 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.

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.

Centered planes in the projective connection space

2020 ◽  
pp. 29-38
Author(s):  
O. O. Belova
Keyword(s):  
Vector Bundle ◽  
Projective Space ◽  
Grassmann Manifold ◽  
Quotient Space ◽  
Differentiable Manifold ◽  
Projective Connection ◽  
Basic Space ◽  
And Topology ◽  
Connection Space ◽  
Dimensional Projective Space

The space of centered planes is considered in the Cartan projec­ti­ve connection space . The space is important because it has con­nec­tion with the Grassmann manifold, which plays an important role in geometry and topology, since it is the basic space of a universal vector bundle. The space is an n-dimensional differentiable manifold with each point of which an n-dimensional projective space containing this point is associated. Thus, the manifold is the base, and the space is the n-dimensional fiber “glued” to the points of the base. A projective space is a quotient space of a linear space with respect to the equivalence (collinearity) of non-zero vectors, that is . The projective space is a manifold of di­men­sion n. In this paper we use the Laptev — Lumiste invariant analytical meth­od of differential geometric studies of the space of centered planes and introduce a fundamental-group connection in the associated bundle . The bundle contains four quotient bundles. It is show that the connection object is a quasi-tensor containing four subquasi-tensors that define connections in the corresponding quotient bundles.

Glued linear connection on surface of the projective space

2020 ◽  
pp. 22-28
Author(s):  
K. V. Bashashina
Keyword(s):  
Projective Space ◽  
Fiber Bundle ◽  
Principal Bundle ◽  
Differential Forms ◽  
Linear Connection ◽  
Projective Connection ◽  
The Given

We consider a surface as a variety of centered planes in a multidi­mensional projective space. A fiber bundle of the linear coframes appears over this manifold. It is important to emphasize the fiber bundle is not the principal bundle. We called it a glued bundle of the linear coframes. A connection is set by the Laptev — Lumiste method in the fiber bundle. The ifferential equations of the connection object components have been found. This leads to a space of the glued linear connection. The expres­sions for a curvature object of the given connection are found in the pa­per. The theorem is proved that the curvature object is a tensor. A condi­tion is found under which the space of the glued linear connection turns into the space of Cartan projective connection. The study uses the Cartan — Laptev method, which is based on cal­culating external differential forms. Moreover, all considerations in the article have a local manner.

scholarly journals On B- Covariant Derivative of First Order for Some Tensors in different Spaces

2021 ◽  
Vol 2 (2) ◽  
pp. 30-37
Author(s):  
Alaa A. Abdallah ◽  
A. A. Navlekar ◽  
Kirtiwant P. Ghadle
Keyword(s):  
Curvature Tensor ◽  
Covariant Derivative ◽  
Torsion Tensor ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
First Order ◽  
Recurrence Property ◽  
Necessary And Sufficient ◽  
The Relationship

In this paper, we study the relationship between Cartan's second curvature tensor $P_{jkh}^{i}$ and $(h) hv-$torsion tensor $C_{jk}^{i}$ in sense of Berwald. Moreover, we discuss the necessary and sufficient condition for some tensors which satisfy a recurrence property in $BC$-$RF_{n}$, $P2$-Like-$BC$-$RF_{n}$, $P^{\ast }$-$BC$-$RF_{n}$ and $P$-reducible-$BC-RF_{n}$.

scholarly journals NATURAL CONNECTION WITH TOTALLY SKEW-SYMMETRIC TORSION ON ALMOST CONTACT MANIFOLDS WITH B-METRIC

2012 ◽  
Vol 09 (05) ◽  
pp. 1250044 ◽  
Author(s):  
MANCHO MANEV
Keyword(s):  
Curvature Tensor ◽  
Torsion Tensor ◽  
Contact Manifolds ◽  
Natural Connection ◽  
Civita Connection ◽  
Levi Civita Connection

A natural connection with totally skew-symmetric torsion on almost contact manifolds with B-metric is constructed. The class of these manifolds, where the considered connection exists, is determined. Some curvature properties for this connection, when the corresponding curvature tensor has the properties of the curvature tensor for the Levi-Civita connection and the torsion tensor is parallel, are obtained.

scholarly journals Derivation of Field Equations in Space with the Geometric Structure Generated by Metric and Torsion

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Nikolay Yaremenko
Keyword(s):  
Curvature Tensor ◽  
Geometric Structure ◽  
Jacobi Identity ◽  
Torsion Tensor ◽  
Variation Principle ◽  
Field Equations ◽  
Metric Tensor ◽  
Fundamental Tensor ◽  
Geodesic Lines ◽  
General Field

This paper is devoted to the derivation of field equations in space with the geometric structure generated by metric and torsion tensors. We also study the geometry of the space generated jointly and agreed on by the metric tensor and the torsion tensor. We showed that in such space the structure of the curvature tensor has special features and for this tensor we obtained analog Ricci-Jacobi identity and evaluated the gap that occurs at the transition from the original to the image and vice versa, in the case of infinitely small contours. We have researched the geodesic lines equation. We introduce the tensor παβ which is similar to the second fundamental tensor of hypersurfaces Yn-1, but the structure of this tensor is substantially different from the case of Riemannian spaces with zero torsion. Then we obtained formulas which characterize the change of vectors in accompanying basis relative to this basis itself. Taking into considerations our results about the structure of such space we derived from the variation principle the general field equations (electromagnetic and gravitational).

Topological aspects of the projective unitary group

1970 ◽  
Vol 68 (1) ◽  
pp. 57-60 ◽  
Author(s):  
D. J. Simms
Keyword(s):  
Hilbert Space ◽  
Topological Group ◽  
Projective Space ◽  
Unitary Group ◽  
Principal Bundle ◽  
Strong Operator Topology ◽  
Complex Hilbert Space ◽  
Unitary Representations ◽  
Strong Operator ◽  
Unitary Transformations

1. Introduction. The group U(H) of unitary transformations of a complex Hilbert space H, endowed with its strong operator topology, is of interest in the study of unitary representations of a topological group. The unitary transformations of H induce a group U(Ĥ) of transformations of the associated projective space Ĥ. The projective unitary group U(Ĥ) with its strong operator topology is used in the study of projective (ray) representations. U(Ĥ) is, as a group, the quotient of U(H) by the subgroup S1 of scalar multiples of the identity. In this paper we prove that the strong operator toplogy of U(Ĥ) is in fact the quotient of the strong operator topology on U(H). This is related to the fact that U(H) is a principal bundle over U(Ĥ) with fibre S.

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