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.

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

GENERALIZED CONNECTIONS AND CHARACTERISTIC CLASSES

1991 ◽  
Vol 02 (05) ◽  
pp. 515-524
Author(s):  
HONG-JONG KIM
Keyword(s):  
Smooth Manifold ◽  
Vector Bundles ◽  
Characteristic Classes ◽  
De Rham Cohomology ◽  
De Rham ◽  
Twisted De Rham Cohomology

We study derivations on a smooth manifold, its twisted de Rham cohomology, generalized connections on vector bundles and their characteristic classes.

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].

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.

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