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.

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 .

Vertical Chern Type Classes on Complex Finsler Bundles

2011 ◽  
Vol 57 (2) ◽  
pp. 377-386
Author(s):  
Cristian Ida
Keyword(s):  
Differential Forms ◽  
Linear Connection ◽  
Curvature Form ◽  
Cohomology Groups ◽  
Finsler Manifolds ◽  
Invariance Properties ◽  
Type Classes

Vertical Chern Type Classes on Complex Finsler BundlesIn the present paper, we define vertical Chern type classes on complex Finsler bundles, as an extension of thev-cohomology groups theory on complex Finsler manifolds. These classes are introduced in a classical way by using closed differential forms with respect to the conjugated vertical differential in terms of the vertical curvature form of Chern-Finsler linear connection. Also, some invariance properties of these classes are studied.

Basic Forms

2020 ◽  
pp. 97-102
Author(s):  
Loring W. Tu
Keyword(s):  
Lie Group ◽  
Principal Bundle ◽  
Differential Forms ◽  
Total Space ◽  
Lie Derivative ◽  
Connected Lie Group

This chapter describes basic forms. On a principal bundle π‎: P → M, the differential forms on P that are pullbacks of forms ω‎ on the base M are called basic forms. The chapter characterizes basic forms in terms of the Lie derivative and interior multiplication. It shows that basic forms on a principal bundle are invariant and horizontal. To understand basic forms better, the chapter considers a simple example. The plane ℝ2 may be viewed as the total space of a principal ℝ-bundle. A connected Lie group is generated by any neighborhood of the identity. This example shows the necessity of the connectedness hypothesis.

Homotopy Quotients and Equivariant Cohomology

2020 ◽  
pp. 29-38
Author(s):  
Loring W. Tu
Keyword(s):  
Topological Group ◽  
Fiber Bundle ◽  
Equivariant Cohomology ◽  
Principal Bundle ◽  
Total Space

This chapter investigates two candidates for equivariant cohomology and explains why it settles on the Borel construction, also called Cartan's mixing construction. Let G be a topological group and M a left G-space. The Borel construction mixes the weakly contractible total space of a principal bundle with the G-space M to produce a homotopy quotient of M. Equivariant cohomology is the cohomology of the homotopy quotient. More generally, given a G-space M, Cartan's mixing construction turns a principal bundle with fiber G into a fiber bundle with fiber M. Cartan's mixing construction fits into the Cartan's mixing diagram, a powerful tool for dealing with equivariant cohomology.

scholarly journals Antisymmetric Paramodular Forms of Weights 2 and 3

2019 ◽  
Vol 2020 (20) ◽  
pp. 6926-6946 ◽  
Author(s):  
Valery Gritsenko ◽  
Cris Poor ◽  
David S Yuen
Keyword(s):  
Projective Space ◽  
Modular Forms ◽  
Differential Forms ◽  
Rational Points ◽  
Siegel Modular Forms ◽  
Abelian Surfaces ◽  
Algebraic Set ◽  
Minimal Weight ◽  
Dimensional Projective Space

Abstract We define an algebraic set in $23$-dimensional projective space whose ${{\mathbb{Q}}}$-rational points correspond to meromorphic, antisymmetric, paramodular Borcherds products. We know two lines inside this algebraic set. Some rational points on these lines give holomorphic Borcherds products and thus construct examples of Siegel modular forms on degree 2 paramodular groups. Weight $3$ examples provide antisymmetric canonical differential forms on Siegel modular three-folds. Weight $2$ is the minimal weight and these examples, via the paramodular conjecture, give evidence for the modularity of some rank 1 abelian surfaces defined over $\mathbb{Q}$.

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.

scholarly journals CLIFFORD VALUED DIFFERENTIAL FORMS, AND SOME ISSUES IN GRAVITATION, ELECTROMAGNETISM AND "UNIFIED" THEORIES

2004 ◽  
Vol 13 (09) ◽  
pp. 1879-1915 ◽  
Author(s):  
W. A. RODRIGUES ◽  
E. CAPELAS DE OLIVEIRA
Keyword(s):  
Lie Algebra ◽  
Differential Forms ◽  
Linear Connection ◽  
Gravitational Theory ◽  
Momentum Conservation ◽  
Unified Field Theory ◽  
Mathematical Methods ◽  
Unified Field ◽  
Theory Of Gravitation ◽  
Local Trivialization

In this paper we show how to describe the general theory of a linear metric compatible connection with the theory of Clifford valued differential forms. This is done by realizing that for each spacetime point the Lie algebra of Clifford bivectors is isomorphic to the Lie algebra of [Formula: see text]. In that way the pullback of the linear connection under a local trivialization of the bundle (i.e., a choice of gauge) is represented by a Clifford valued 1-form. That observation makes it possible to realize immediately that Einstein's gravitational theory can be formulated in a way which is similar to a [Formula: see text] gauge theory. Such a theory is compared with other interesting mathematical formulations of Einstein's theory, and particularly with a supposedly "unified" field theory of gravitation and electromagnetism proposed by M. Sachs. We show that his identification of Maxwell equations within his formalism is not a valid one. Also, taking profit of the mathematical methods introduced in the paper we investigate a very polemical issue in Einstein gravitational theory, namely the problem of the 'energy–momentum' conservation. We show that many statements appearing in the literature are confusing or even wrong.

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.

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.

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