COHOMOLOGY OF LAGRANGE COMPLEXES INVARIANT UNDER PSEUDOGROUPS OF LOCAL TRANSFORMATIONS

2007 ◽  
Vol 04 (04) ◽  
pp. 669-705 ◽  
Author(s):  
ANDREA SPIRO
Keyword(s):  
Inverse Problem ◽  
Calculus Of Variations ◽  
Configuration Space ◽  
Cohomology Class ◽  
Cohomology Groups ◽  
Lagrange Equations ◽  
De Rham Cohomology ◽  
Base Manifold ◽  
De Rham

The inverse problem of the Calculus of Variations for Lagrangians and Euler–Lagrange equations invariant under a pseudogroup [Formula: see text] of local transformations of the base manifold is considered. Exploiting some ideas of Krupka, a theorem is proved showing that, if the configuration space consists of sections of tensor bundles or of local maps of a manifold into another, then such inverse problem is solvable whenever a certain cohomology class of [Formula: see text]-invariant forms on the configuration space is vanishing. In addition, for a few pseudogroups, the cohomology groups considered in the main result are explicitly determined in terms of the de Rham cohomology of the configuration space.

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 NONTRIVIAL CLASSES IN H*(Imb(S1,ℝn)) FROM NONTRIVALENT GRAPH COCYCLES

2004 ◽  
Vol 01 (05) ◽  
pp. 639-650 ◽  
Author(s):  
RICCARDO LONGONI
Keyword(s):  
Linear Combination ◽  
Cohomology Class ◽  
Feynman Diagrams ◽  
De Rham Cohomology ◽  
De Rham

We construct nontrivial cohomology classes of the space Imb (S1,ℝn) of imbeddings of the circle into ℝn by means of Feynman diagrams. More precisely, starting from a suitable linear combination of nontrivalent diagrams, we construct, for every even number n≥4, a de Rham cohomology class on Imb (S1,ℝn). We prove nontriviality of these classes by evaluation on the dual cycles.

scholarly journals A Study of Doubly Warped Product Immersions in a Nearly Trans-Sasakian Manifold with Slant Factor

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ali H. Alkhaldi ◽  
Aliya Naaz Siddiqui ◽  
Kamran Ahmad ◽  
Akram Ali
Keyword(s):  
Cohomology Class ◽  
Warped Product ◽  
Sasakian Manifold ◽  
Sasakian Manifolds ◽  
De Rham Cohomology ◽  
De Rham

In this article, we discuss the de Rham cohomology class for bislant submanifolds in nearly trans-Sasakian manifolds. Moreover, we give a classification of warped product bislant submanifolds in nearly trans-Sasakian manifolds with some nontrivial examples in the support. Next, it is of great interest to prove that there does not exist any doubly warped product bislant submanifolds other than warped product bislant submanifolds in nearly trans-Sasakian manifolds. Some immediate consequences are also obtained.

scholarly journals Representing de Rham cohomology classes on an open Riemann surface by holomorphic forms

2017 ◽  
Vol 28 (09) ◽  
pp. 1740004 ◽  
Author(s):  
Antonio Alarcón ◽  
Finnur Lárusson
Keyword(s):  
Riemann Surface ◽  
Convex Hull ◽  
Minimal Surface ◽  
Cohomology Class ◽  
Holomorphic Maps ◽  
De Rham Cohomology ◽  
Surface Theory ◽  
Open Riemann Surface ◽  
De Rham ◽  
Smooth Locus

Let [Formula: see text] be a connected open Riemann surface. Let [Formula: see text] be an Oka domain in the smooth locus of an analytic subvariety of [Formula: see text], [Formula: see text], such that the convex hull of [Formula: see text] is all of [Formula: see text]. Let [Formula: see text] be the space of nondegenerate holomorphic maps [Formula: see text]. Take a holomorphic 1-form [Formula: see text] on [Formula: see text], not identically zero, and let [Formula: see text] send a map [Formula: see text] to the cohomology class of [Formula: see text]. Our main theorem states that [Formula: see text] is a Serre fibration. This result subsumes the 1971 theorem of Kusunoki and Sainouchi that both the periods and the divisor of a holomorphic form on [Formula: see text] can be prescribed arbitrarily. It also subsumes two parametric h-principles in minimal surface theory proved by Forstnerič and Lárusson in 2016.

A toral configuration space and regular semisimple conjugacy classes

1995 ◽  
Vol 118 (1) ◽  
pp. 105-113 ◽  
Author(s):  
G. I. Lehrer
Keyword(s):  
Topological Space ◽  
Symmetric Group ◽  
Configuration Space ◽  
Braid Group ◽  
Cohomology Ring ◽  
General Setting ◽  
De Rham Cohomology ◽  
Pure Braid Group ◽  
De Rham ◽  
Image Position

For any topological space X and integer n ≥ 1, denote by Cn(X) the configuration spaceThe symmetric group Sn acts by permuting coordinates on Cn(X) and we are concerned in this note with the induced graded representation of Sn on the cohomology space H*(Cn(X)) = ⊕iHi (Cn(X), ℂ), where Hi denotes (singular or de Rham) cohomology. When X = ℂ, Cn(X) is a K(π, 1) space, where π is the n-string pure braid group (cf. [3]). The corresponding representation of Sn in this case was determined in [5], using the fact that Cn(C) is a hyperplane complement and a presentation of its cohomology ring appears in [1] and in a more general setting, in [8] (see also [2]).

scholarly journals Twisted de Rham Cohomology Groups of Logarithmic Forms

1997 ◽  
Vol 128 (1) ◽  
pp. 119-152 ◽  
Author(s):  
Kazuhiko Aomoto ◽  
Michitake Kita ◽  
Peter Orlik ◽  
Hiroaki Terao
Keyword(s):  
Cohomology Groups ◽  
De Rham Cohomology ◽  
De Rham ◽  
Twisted De Rham Cohomology

scholarly journals Study on De Rham Cohomology Algebra of Manifolds

2016 ◽  
Vol 64 (2) ◽  
pp. 109-113
Author(s):  
Saraban Tahora ◽  
Khondokar M Ahmed
Keyword(s):  
Cohomology Class ◽  
Homology Class ◽  
Cohomology Algebra ◽  
Graded Algebra ◽  
Exterior Derivative ◽  
De Rham Cohomology ◽  
Singular Homology ◽  
Orientable Manifold ◽  
De Rham ◽  
Smooth Map

In the present paper some aspects of exterior derivative, graded algebra, cohomology algebra, de Rham cohomology algebra, singular homology, cohomology class are studied. Graded subspace, smooth map, a singular P- - simplex in a manifold M, oriented n- manifold M, the space of P- cycles and P- boundaries, Pth singular homology and homology class are treated in our paper. A theorem 3.03 is established which is related to orientable manifold. Dhaka Univ. J. Sci. 64(2): 109-113, 2016 (July)

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