De Rham Theorem Research Articles

Algebraic de Rham theorem and Baker-Akhiezer function

For the case of algebraic curves (compact Riemann surfaces), it is shown that de Rham cohomology group $H^1_{\mathrm{dR}}(X,\mathbb{C})$ of a genus $g$ of the Riemann surface $X$ has a natural structure of a symplectic vector space. Every choice of a non-special effective divisor $D$ of degree $g$ on $X$ defines a symplectic basis of $H^1_{\mathrm{dR}}(X,\mathbb{C})$ consisting of holomorphic differentials and differentials of the second kind with poles on $D$. This result, which is the algebraic de Rham theorem, is used to describe the tangent space to Picard and Jacobian varieties of $X$ in terms of differentials of the second kind, and to define a natural vector fields on the Jacobian of the curve $X$ that move points of the divisor $D$. In terms of the Lax formalism on algebraic curves, these vector fields correspond to the Dubrovin equations in the theory of integrable systems, and the Baker-Akhierzer function is naturally obtained by the integration along the integral curves.

Differentiable stratified groupoids and a de Rham theorem for inertia spaces

We introduce the notions of differentiable groupoids and differentiable stratified groupoids, generalizations of Lie groupoids in which the spaces of objects and arrows have the structures of differentiable spaces, respectively differentiable stratified spaces, compatible with the groupoid structure. After studying basic properties of these groupoids including Morita equivalence, we prove a de Rham theorem for proper locally contractible differentiable stratified groupoids. We then focus on the study of the inertia groupoid associated to a proper Lie groupoid. We show that the loop and the inertia space of a proper Lie groupoid can be endowed with a natural Whitney (b)-regular stratification, which we call the orbit Cartan type stratification. Endowed with this stratification, the inertia groupoid of a proper Lie groupoid becomes a locally contractible differentiable stratified groupoid.

A generalized de Rham Theorem for Intersection Spaces

A generalized de Rham Theorem for Intersection Spaces

Finite Element Systems for Vector Bundles: Elasticity and Curvature

We develop a theory of finite element systems, for the purpose of discretizing sections of vector bundles, in particular those arising in the theory of elasticity. In the presence of curvature, we prove a discrete Bianchi identity. In the flat case, we prove a de Rham theorem on cohomology groups. We check that some known mixed finite elements for the stress–displacement formulation of elasticity fit our framework. We also define, in dimension two, the first conforming finite element spaces of metrics with good linearized curvature, corresponding to strain tensors with Saint-Venant compatibility conditions. Cochains with coefficients in rigid motions are given a key role in relating continuous and discrete elasticity complexes.

Flat forms in metric spaces

In a metric space, we define flat forms by means of tuples of Lipschitz functions multiplied by Borel measurable functions, and use them to represent flat cochains. The representation, which extends Wolfe's theorem to metric spaces, is functorial on the category of metric spaces and Lipschitz maps. It provides flat cochains and flat chains with well-behaved cup and cap products. On compact Lipschitz manifolds, the cohomology of flat cochains is naturally isomorphic to the Čech cohomology with real coefficients — a version of De Rham's theorem.

On a version of the de Rham theorem and an application to the Maxwell–Stokes type problem

In this paper, we consider a version of the de Rham theorem that clarifies the additional information on the trace of the potential. By using this result, we can improve the existence of a weak solution to a Maxwell–Stokes type system in a multi-connected domain with holes.

On the de Rham Theorem and an Application to the Maxwell–Stokes Type Problem

In this paper, we derive an $$L^{p}$$ version of the de Rham theorem. The key is an $$L^{p}$$ version of the Necas inequality. Using this result and the variational method, we show the existence of a solution to the Maxwell–Stokes type system.

On the curl operator and some characterizations of matrix fields in Lipschitz domains

As well-known, De Rham's Theorem is a classical way to characterize vector fields as the gradient of the scalar fields, it is a tool of great importance in the theory of fluids mechanic. The first aim of this paper is to provide a useful rotational version of this theorem to establish several results on boundary value problems in the field of electromagnetism. Gurtin presented the completeness of Beltrami's representation in smooth domains for smooth symmetric matrix. The extension of Beltrami's completeness was obtained by Geymonat and Krasucki when the domain Ω is only Lipschitz and for symmetric matrix in Ls2(Ω). Our second aim is to present new extensions of Beltrami's completeness results on Lipschitz domains, firstly in the case where the data are in Ds(Ω) and secondly when the data are in Wsm,r(Ω). The third objective is to extend Saint-Venant's Theorem to the case of distributions.

Simplicial cochain algebras for diffeological spaces

The original de Rham cohomology due to Souriau and the singular cohomology in diffeology are not isomorphic to each other in general. This manuscript introduces a singular de Rham complex endowed with an integration map into the singular cochain complex which gives the de Rham theorem for every diffeological space. It is also proved that a morphism called the factor map from the original de Rham complex to the new one is a quasi-isomorphism for a manifold and, more general, a space with singularities. Moreover, Chen’s iterated integrals are considered in a diffeological framework. As a consequence, we deduce that the bar complex of the original de Rham complex of a simply-connected diffeological space is quasi-isomorphic to the singular de Rham complex of the diffeological free loop space provided the factor map for the underlying diffeological space is a quasi-isomorphism. The process for proving the assertion yields the Leray–Serre spectral sequence and the Eilenberg–Moore spectral sequence in diffeology.

Inner differentiability and differential forms on tangentially locally linearly independent sets

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.

De Rham theorem for Whitney functions

Let M be a real analytic manifold, F a bounded complex of constructible sheaves. We show that the Whitney–de Rham complex associated to F is quasi-isomorphic to F.

Smooth covers on symplectic manifolds

In this article, we first introduce the notion of a continuous cover of a manifold parametrised by any compact manifold T endowed with a mass 1 volume-form. We prove that any such cover admits a partition of unity where the sum is replaced by integrals. When the cover is smooth, we then generalize Polterovich’s notion of Poisson non-commutativity to such a context in order to get a more natural definition of non-commutativity and to be in a position where one can compare various invariants of symplectic manifolds. The main theorem of this article states that the discrete Poisson bracket invariant of Polterovich is equal to our smooth version of it, as it does not depend on the nature or dimension of the parameter space T. As a consequence, the Poisson-bracket invariant of a symplectic manifold can be computed either in the discrete category or in the smooth one, that is to say either by summing or integrating. The latter is in general more amenable to calculations, so that, in some sense, our result is in the spirit of the De Rham theorem equating simplicial cohomology and De Rham cohomology. We finally study the Poisson-bracket invariant associated to coverings by symplectic balls of capacity c, exhibiting some of its properties as the capacity c varies. We end with some positive and negative speculations on the relation between uncertainty phase transitions and critical values of the Poisson bracket, which was the motivation behind this article.

De Rham theory and cocycles of cubical sets from smooth quandles

We show a de Rham theorem for cubical manifolds, and study rational homotopy type of the classifying spaces of smooth quandles. We also show that secondary characteristic classes in [8, 9] produce cocycles of quandles.

Multiplicative de Rham Theorems for Relative and Intersection Space Cohomology

We construct an explicit de Rham isomorphism relating the cohomology rings of Banagl's de Rham and spatial approach to intersection space cohomology for stratified pseudomanifolds with isolated singularities. Intersection space (co-)homology is a modified (co-)homology theory extending Poincar\'e Duality to stratified pseudomanifolds. The novelty of our result compared to the de Rham isomorphism given previously by Banagl is, that we indeed have an isomorphism of rings and not just of graded vector spaces. We also provide a proof of the de Rham Theorem for cohomology rings of pairs of smooth manifolds which we use in the proof of our main result.

Proton spin: A topological invariant

Proton spin problem is given a new perspective with the proposition that spin is a topological invariant represented by a de Rham 3-period. The idea is developed generalizing Finkelstein–Rubinstein theory for Skyrmions/kinks to topological defects, and using non-Abelian de Rham theorems. Two kinds of de Rham theorems are discussed applicable to matrix-valued differential forms, and traces. Physical and mathematical interpretations of de Rham periods are presented. It is suggested that Wilson lines and loop operators probe the local properties of the topology, and spin as a topological invariant in pDIS measurements could appear with any value from 0 to [Formula: see text], i.e. proton spin decomposition has no meaning in this approach.

Global martingale solution to the stochastic nonhomogeneous magnetohydrodynamics system

We study the three-dimensional stochastic nonhomogeneous magnetohydrodynamics system with random external forces that involve feedback, i.e., multiplicative noise, and are non-Lipschitz. We prove the existence of a global martingale solution via a semi-Galerkin approximation scheme with stochastic calculus and applications of Prokhorov's and Skorokhod's theorems. Furthermore, using de Rham's theorem for processes, we prove the existence of the pressure term.

Global martingale solution for the stochastic Boussinesq system with zero dissipation

ABSTRACTWe study the two-dimensional stochastic Boussinesq system with zero dissipation and multiplicative noise. We show the existence of a martingale solution by a priori estimates using stochastic calculus, and applications of Prokhorov's, Skorokhod's, and martingale representation theorems. Due to the lack of dissipation, the proof requires higher regularity estimates, taking advantage of the structure of the nonlinear term. Moreover, we obtain the existence of the pressure term via an application of de Rham's theorem for processes.

A nonpercolation problem for the stationary Navier-Stokes equations

The solvability of a nonpercolation boundary problem for the stationary Navier-Stokes equations is proved. The key points of the proof are analogues of the Friedrichs inequality and the de Rham theorem adequate for nonpercolation conditions.

Boundary Value Problem for Stationary Stokes Equations with Impermeability Boundary Condition

We propose two approaches to the study of the boundary value problem for the stationary Stokes equations with impermeability boundary condition. The first approach is classical and is based on a Friedrichs type inequality and a variant of the de Rham theorem. The second approach is based on solving the boundary value problem with the impermeability condition for the system of Poisson equations and decomposition of a Sobolev space into the sum of solenoidal and potential subspaces. We also study the gradient-divergence boundary value problem with impermeability boundary condition and establish the corresponding Ladyzhenskaya–Babushka–Brezzi inequality.

Stratifications of Inertia Spaces of Compact Lie Group Actions

We study the topology of the inertia space of a smooth $G$-manifold $M$ where $G$ is a compact Lie group. We construct an explicit Whitney stratification of the inertia space, demonstrating that the inertia space is a triangulable differentiable stratified space. In addition, we demonstrate a de Rham theorem for differential forms defined on the inertia space with respect to this stratification.