scholarly journals Inner differentiability and differential forms on tangentially locally linearly independent sets

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 365-372
Author(s):  
Aneta Velkoska ◽  
Zoran Misajleski
Keyword(s):  
Euclidean Space ◽  
Differential Forms ◽  
Differentiable Manifold ◽  
Independent Sets ◽  
Natural Isomorphism ◽  
De Rham Cohomology ◽  
Linearly Independent ◽  
Singular Cohomology ◽  
Real Functions ◽  
De Rham

The de Rham theorem gives a natural isomorphism between De Rham cohomology and singular cohomology on a paracompact differentiable manifold. We proved this theorem on a wider family of subsets of Euclidean space, on which we can define inner differentiability. Here we define this family of sets called tangentially locally linearly independent sets, propose inner differentiability on them, postulate usual properties of differentiable real functions and show that the integration over sets that are wider than manifolds is possible.

scholarly journals The de Rham Cohomology through Hilbert Space Methods

2019 ◽  
pp. 455-465
Author(s):  
Ihsane Malass ◽  
Nikolai Tarkhanov
Keyword(s):  
Hilbert Space ◽  
Elliptic Boundary ◽  
Differential Forms ◽  
Adjoint Operator ◽  
Lagrange Equations ◽  
De Rham Cohomology ◽  
Elliptic Boundary Value ◽  
De Rham ◽  
Canonical Representations ◽  
The Neumann Problem

We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler- Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer

scholarly journals The de Rham theorem for the noncommutative complex of Cenkl and Porter

2002 ◽  
Vol 30 (11) ◽  
pp. 667-696 ◽  
Author(s):  
Luis Fernando Mejias
Keyword(s):  
Finite Type ◽  
Differential Forms ◽  
Natural Isomorphism ◽  
Graded Algebra ◽  
Differential Graded Algebra ◽  
Noncommutative Version ◽  
Simplicial Set ◽  
De Rham ◽  
Poincaré Lemma

We use noncommutative differential forms (which were first introduced by Connes) to construct a noncommutative version of the complex of Cenkl and PorterΩ∗,∗(X)for a simplicial setX. The algebraΩ∗,∗(X)is a differential graded algebra with a filtrationΩ∗,q(X)⊂Ω∗,q+1(X), such thatΩ∗,q(X)is aℚq-module, whereℚ0=ℚ1=ℤandℚq=ℤ[1/2,…,1/q]forq>1. Then we use noncommutative versions of the Poincaré lemma and Stokes' theorem to prove the noncommutative tame de Rham theorem: ifXis a simplicial set of finite type, then for eachq≥1and anyℚq-moduleM, integration of forms induces a natural isomorphism ofℚq-modulesI:Hi(Ω∗,q(X),M)→Hi(X;M)for alli≥0. Next, we introduce a complex of noncommutative tame de Rham currentsΩ∗,∗(X)and we prove the noncommutative tame de Rham theorem for homology: ifXis a simplicial set of finite type, then for eachq≥1and anyℚq-moduleM, there is a natural isomorphism ofℚq-modulesI:Hi(X;M)→Hi(Ω∗,q(X),M)for alli≥0.

Variations of Integrals in Diffeology

2013 ◽  
Vol 65 (6) ◽  
pp. 1255-1286
Author(s):  
Patrick Iglesias-Zemmour
Keyword(s):  
Differential Forms ◽  
Moment Map ◽  
De Rham Cohomology ◽  
Homotopy Operator ◽  
Chain Homotopy ◽  
De Rham ◽  
A Chain ◽  
The Moment

AbstractWe establish a formula for the variation of integrals of differential forms on cubic chains in the context of diffeological spaces. Then we establish the diffeological version of Stokes’ theorem, and we apply that to get the diffeological variant of the Cartan–Lie formula. Still in the context of Cartan–De Rham calculus in diffeology, we construct a chain-homotopy operator K, and we apply it here to get the homotopic invariance of De Rham cohomology for diffeological spaces. This is the chain-homotopy operator that is used in symplectic diffeology to construct the moment map.

A DE RHAM ISOMORPHISM IN SINGULAR COHOMOLOGY AND STOKES THEOREM FOR STRATIFOLDS

2005 ◽  
Vol 02 (01) ◽  
pp. 63-81 ◽  
Author(s):  
CHRISTIAN-OLIVER EWALD
Keyword(s):  
Geometric Construction ◽  
Theoretical Physics ◽  
Smooth Manifolds ◽  
De Rham Cohomology ◽  
Singular Cohomology ◽  
De Rham

We study the de Rham cohomology of a class of spaces with singularities which are called stratifolds. Such spaces occur in mathematical and theoretical physics. We generalize two classical results from the analysis of smooth manifolds, with outstanding applications in physics, to the class of stratifolds. The first one is Stokes theorem, the second one is the de Rham theorem which states that the de Rham cohomology of a stratifold is isomorphic to its singular cohomology with coefficients in ℝ. In fact, we give an explicit geometric construction of this isomorphism, given by integrating forms over stratifolds.

scholarly journals On twisted de Rham cohomology

1997 ◽  
Vol 146 ◽  
pp. 55-81 ◽  
Author(s):  
Alan Adolphson ◽  
Steven Sperber
Keyword(s):  
Differential Forms ◽  
Exterior Derivative ◽  
De Rham Cohomology ◽  
De Rham ◽  
Twisted De Rham Cohomology

Abstract.Consider the complex of differential forms on an open affine subvariety U of AN with differential where d is the usual exterior derivative and ø is a fixed 1-form on U. For certain U and ø, we compute the cohomology of this complex.

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