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.

ON PROJECTIVELY RELATED WARPED PRODUCT FINSLER MANIFOLDS

2011 ◽  
Vol 08 (05) ◽  
pp. 953-967 ◽  
Author(s):  
M. M. REZAII ◽  
Y. ALIPOUR-FAKHRI
Keyword(s):  
Warped Product ◽  
Sufficient Conditions ◽  
Linear Connection ◽  
Necessary And Sufficient Conditions ◽  
Finsler Manifolds ◽  
Non Linear ◽  
Necessary And Sufficient

Let 𝔽1 = (M1,F1) and 𝔽2 = (M2,F2) be two Finsler manifolds and let M = M1 × M2 and S is a spray in M. Also 𝔽 = (M1 × f M2, F) is a warped product Finsler manifolds, such that the function f : M1 → ℝ+ is not constant. In this paper, we define a non-linear connection on warped product 𝔽, and finally, we have presented some necessary and sufficient conditions under which the spray manifold (M1 × M2, S) is projectively equivalent to the warped product Finsler manifolds (M1 × f M2, F).

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.

Residue: A Geometric Construction

2002 ◽  
Vol 45 (2) ◽  
pp. 284-293 ◽  
Author(s):  
Fernando Sancho de Salas
Keyword(s):  
Geometric Structure ◽  
Differential Forms ◽  
Geometric Construction ◽  
Residue Theorem ◽  
Cohomology Groups ◽  
Local Coordinates ◽  
New Construction

AbstractA new construction of the ordinary residue of differential forms is given. This construction is intrinsic, i.e., it is defined without local coordinates, and it is geometric: it is constructed out of the geometric structure of the local and global cohomology groups of the differentials. The Residue Theorem and the local calculation then follow from geometric reasons.

VERTICAL TANGENTIAL INVARIANTS ON SOME FOLIATED LAGRANGE SPACES

2013 ◽  
Vol 10 (04) ◽  
pp. 1320002
Author(s):  
CRISTIAN IDA
Keyword(s):  
Tangent Bundle ◽  
Differential Forms ◽  
Characteristic Classes ◽  
Exterior Derivative ◽  
Cohomology Groups ◽  
Foliated Manifold ◽  
Smooth Complex ◽  
De Rham ◽  
Poincaré Lemma ◽  
Natural Decomposition

In this paper we consider a decomposition of tangentially differential forms with respect to the lifted foliation [Formula: see text] to the tangent bundle of a Lagrange space [Formula: see text] endowed with a regular foliation [Formula: see text]. First, starting from a natural decomposition of the tangential exterior derivative along the leaves of [Formula: see text], we define some vertical tangential cohomology groups of the foliated manifold [Formula: see text], we prove a Poincaré lemma for the vertical tangential derivative and we obtain a de Rham theorem for this cohomology. Next, in a classical way, we construct vertical tangential characteristic classes of tangentially smooth complex bundles over the foliated manifold [Formula: see text].

scholarly journals DEFORMATION THEORY OF THE CHOW GROUP OF ZERO CYCLES

2020 ◽  
Vol 71 (2) ◽  
pp. 677-676
Author(s):  
Morten Lüders
Keyword(s):  
Valuation Ring ◽  
Deformation Theory ◽  
Differential Forms ◽  
Discrete Valuation Ring ◽  
Chow Group ◽  
Cohomology Groups ◽  
Lefschetz Theorems ◽  
Special Fibre

Abstract We study the deformations of the Chow group of zerocycles of the special fibre of a smooth scheme over a Henselian discrete valuation ring. Our main tools are Bloch’s formula and differential forms. As a corollary we get an algebraization theorem for thickened zero cycles previously obtained using idelic techniques. In the course of the proof we develop moving lemmata and Lefschetz theorems for cohomology groups with coefficients in differential forms.

scholarly journals LINEAR CONNECTIONS ON THE TWO-PARAMETER QUANTUM PLANE

1996 ◽  
Vol 08 (08) ◽  
pp. 1055-1060 ◽  
Author(s):  
YVON GEORGELIN ◽  
JEAN-CHRISTOPHE WALLET ◽  
THIERRY MASSON
Keyword(s):  
Differential Forms ◽  
Linear Connection ◽  
Quantum Plane ◽  
Specific Relation ◽  
Linear Connections ◽  
Definition Of ◽  
Two Parameters ◽  
Two Parameter

We apply a recently proposed definition of a linear connection in non-commutative geometry based on the natural bimodule structure of the algebra of differential forms to the case of the two-parameter quantum plane. We find that there exists a non-trivial family of linear connections only when the two parameters obey a specific relation.

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.

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