scholarly journals Exterior Algebra with Differential Forms on Manifolds

2012 ◽  
Vol 60 (2) ◽  
pp. 247-252
Author(s):  
Md. Showkat Ali ◽  
K.M. Ahmed ◽  
M.R. Khan ◽  
Md. Mirazul Islam
Keyword(s):  
Differential Geometry ◽  
Differential Equations ◽  
Differential Forms ◽  
Exterior Algebra ◽  
Linear Spaces ◽  
Exterior Differentiation

The concept of an exterior algebra was originally introduced by H. Grassman for the purpose of studying linear spaces. Subsequently Elie Cartan developed the theory of exterior differentiation and successfully applied it to the study of differential geometry [8], [9] or differential equations. More recently, exterior algebra has become powerful and irreplaceable tools in the study of differential manifolds with differential forms and we develop theorems on exterior algebra with examples.DOI: http://dx.doi.org/10.3329/dujs.v60i2.11528 Dhaka Univ. J. Sci. 60(2): 247-252, 2012 (July)

scholarly journals Differential forms with values in groups

1982 ◽  
Vol 25 (3) ◽  
pp. 357-386 ◽  
Author(s):  
Anders Kock
Keyword(s):  
Differential Geometry ◽  
Lie Algebra ◽  
Classical Theory ◽  
Lie Group ◽  
Differential Form ◽  
Differential Forms ◽  
Synthetic Differential Geometry ◽  
Exterior Differentiation ◽  
Group Object

In the context of synthetic differential geometry, we present a notion of differential form with values in a group object, typically a Lie group or the group of all diffeomorphisms of a manifold. Natural geometric examples of such forms and the role of their exterior differentiation is given. The main result is a comparison with the classical theory of Lie algebra valued forms.

Equivalence and symmetries of second-order differential equations

2008 ◽  
Vol 58 (5) ◽  
Author(s):  
V. Tryhuk ◽  
O. Dlouhý
Keyword(s):  
Differential Geometry ◽  
Differential Equations ◽  
Differential Forms ◽  
Second Order ◽  
Fundamental Properties ◽  
Second Order Differential Equations ◽  
The Common ◽  
Differential Equations With Deviations ◽  
Direct Calculations

AbstractIn this article we investigate the equivalence of underdetermined differential equations and differential equations with deviations of second order with respect to the pseudogroup of transformations $$ \bar x $$ = φ(x), ȳ = ȳ($$ \bar x $$) = L(x) + y(x), $$ \bar z $$ = $$ \bar z $$($$ \bar x $$) = M(x) + z(x). Our main aim is to determine such equations that admit a large pseudogroup of symmetries. Instead the common direct calculations, we use some more advanced tools from differential geometry, however, our exposition is self-contained and only the most fundamental properties of differential forms are employed.

scholarly journals The diagrammatic coaction beyond one loop

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Samuel Abreu ◽  
Ruth Britto ◽  
Claude Duhr ◽  
Einan Gardi ◽  
James Matthew
Keyword(s):  
Differential Equations ◽  
Hypergeometric Functions ◽  
Differential Forms ◽  
Feynman Graph ◽  
Loop Order ◽  
Feynman Integrals ◽  
Mathematical Properties ◽  
Multiple Polylogarithms ◽  
Complete Set ◽  
General Tool

Abstract The diagrammatic coaction maps any given Feynman graph into pairs of graphs and cut graphs such that, conjecturally, when these graphs are replaced by the corresponding Feynman integrals one obtains a coaction on the respective functions. The coaction on the functions is constructed by pairing a basis of differential forms, corresponding to master integrals, with a basis of integration contours, corresponding to independent cut integrals. At one loop, a general diagrammatic coaction was established using dimensional regularisation, which may be realised in terms of a global coaction on hypergeometric functions, or equivalently, order by order in the ϵ expansion, via a local coaction on multiple polylogarithms. The present paper takes the first steps in generalising the diagrammatic coaction beyond one loop. We first establish general properties that govern the diagrammatic coaction at any loop order. We then focus on examples of two-loop topologies for which all integrals expand into polylogarithms. In each case we determine bases of master integrals and cuts in terms of hypergeometric functions, and then use the global coaction to establish the diagrammatic coaction of all master integrals in the topology. The diagrammatic coaction encodes the complete set of discontinuities of Feynman integrals, as well as the differential equations they satisfy, providing a general tool to understand their physical and mathematical properties.

Differential forms on a Kähler manifold

1951 ◽  
Vol 47 (3) ◽  
pp. 504-517 ◽  
Author(s):  
W. V. D. Hodge
Keyword(s):  
Differential Geometry ◽  
General Theory ◽  
Complex Manifold ◽  
Differential Forms ◽  
Kähler Manifold ◽  
Systematic Account ◽  
General Plan ◽  
Kahler Manifold ◽  
Frequent Use ◽  
Compact Kähler Manifold

While a number of special properties of differential forms on a Kähler manifold have been mentioned in the literature on complex manifolds, no systematic account has yet been given of the theory of differential forms on a compact Kähler manifold. The purpose of this paper is to show how a general theory of these forms can be developed. It follows the general plan of de Rham's paper (2) on differential forms on real manifolds, and frequent use will be made of results contained in that paper. For convenience we begin by giving a brief account of the theory of complex tensors on a complex manifold, and of the differential geometry associated with a Hermitian, and in particular a Kählerian, metric on such a manifold.

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