Rham Theorem Research Articles

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 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.

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.

Holonomies for connections with values in $L_\infty$-algebras

Given a flat connection on a manifold with values in a filtered L-infinity-algebra, we construct a morphism of coalgebras that generalizes the holonomies of flat connections with values in Lie algebras. The construction is based on Gugenheim's A-infinity version of de Rham's theorem, which in turn is based on Chen's iterated integrals. Finally, we discuss examples related to the geometry of configuration spaces of points in Euclidean space, and to generalizations of the holonomy representations of braid groups.

ℓ∞-COHOMOLOGY AND METABOLICITY OF NEGATIVELY CURVED COMPLEXES

We prove the analog of de Rham's theorem for ℓ∞-cohomology of the universal cover of a finite simplicial complex. A sufficient criterion is given for linearity of isoperimetric functions for filling cycles of any positive dimension over ℝ. This implies the linear higher dimensional isoperimetric inequalities for the fundamental groups of finite negatively curved complexes and of closed negatively curved manifolds. Also, these groups are ℝ-metabolic.

The Geometry of Physics: An Introduction

Preface Part I. Manifolds, Tensors and Exterior Forms: 1. Manifolds and vector fields 2. Tensors and exterior forms 3. Integration of differential forms 4. The Lie derivative 5. The Poincare Lemma and potentials 6. Holonomic and non-holonomic constraints Part II. Geometry and Topology: 7. R3 and Minkowski space 8. The geometry of surfaces in R3 9. Covariant differentiation and curvature 10. Geodesics 11. Relativity, tensors, and curvature 12. Curvature and topology: Synge's theorem 13. Betti numbers and De Rham's theorem 14. Harmonic forms Part III. Lie Groups, Bundles and Chern Forms: 15. Lie groups 16. Vector bundles in geometry and physics 17. Fiber bundles, Gauss-Bonnet, and topological quantization 18. Connections and associated bundles 19. The Dirac equation 20. Yang-Mills fields 21. Betti numbers and covering spaces 22. Chern forms and homotopy groups Appendix: forms in continuum mechanics.

Monopoles, gauge fields and de Rham's theorems

The topology assumed by most authors for a spacelike hypersurface in a spacetime containing a monopole is generally ; save for the surface isolating the monopole, this space is unbounded. For such a topology, a consistency relation of de Rham's theorems shows that a single isolated monopole cannot exist. Monopoles, with charge , if they exist at all, must occur in pairs having opposite magnetic charge. An extension of de Rham's theorems to non-Abelian monopoles which are generalizations of Dirac monopoles (those characterized by , the fundamental group of the gauge group G) is made using the definition of an ordered integral of a path-dependent curvature over a surface. This integral is similar to that found in the non-Abelian Stokes theorem. The implications of de Rham's theorems for non-Abelian monopoles are shown to be similar to the Abelian case.

Magnetic energy, multiply connected domains, and force-free fields.

This paper addresses the question of magnetic energy in mutiply connected domains. It is shown that the magnetic energy must in general include a boundary term that is usually assumed to vanish. The physical interpretation of this term is discussed in terms of de Rham's theorems.

On De Rham's theorem in synthetic differential geometry

In classical differential geometry the integration morphism ʃ:ω p(M) → S p(M) is a morphism of complexes inducing an isomorphism in homology. In the synthetic case the two complexes are in fact isomorphic. We exhibit a cochain morphism D: S p ( M) → ω p ( M), which is an inverse for ʃ.

Abstract Riemannian stratifications

RIEMANNIAN STRATIFICATIONS GIACOMO MONTI BRAGADIN Vol. 133, No. 1 March 1988 PACIFIC JOURNAL OF MATHEMATICS Vol. 133, No. 1, 1988 ABSTRACT RIEMANNIAN STRATIFICATIONSRIEMANNIAN STRATIFICATIONS GIACOMO MONTI BRAGADIN Let A be an abstract stratification. Assume that every stratum X is a riemannian manifold. We first give some conditions, under which it is possible to endow A with a suitable metric extending a well known technique of riemannian manifolds. Then stronger conditions are introduced in order that the metric space A becomes a G-space. An abstract stratification with all the above conditions is called an abstract riemannian stratification. Whitney stratifications are, in a natural way, riemannian. Introduction. An abstract stratification is the disjoint union of a locally finite set of smooth manifolds, the strata, glued together according to certain rules (see [9], [13], [15], [16]). The basic tools of differential calculus, smooth functions, vector fields and differential forms, can be also defined for an abstract stratification by means of collections of tools. Classical results can be extended to abstract stratifications (e.g. de Rham's theorems, see [15], [8]) in this way. On the other hand a controlled approach seems to be useless when we take into account a riemannian structure: control conditions on the metric of the strata are too strict (bundle-like metrics, see [11], are not controlled in general). The purpose of this paper is to obtain weaker conditions in order that riemannian metrics of the strata glue together in an appropriate manner. In §1 sufficient conditions are given for extending the fundamental theorem of the riemannian geometry; in §2 a definition of compatibility among the riemannian structures of the strata is given and connections with the structure of G-space are studied. It is shown that the axioms of finite compactness, convexity, local prolongability are satisfied in mild hypotheses, whereas local uniqueness and existence of geodesic coordinates must be required (the former is an open question even in manifolds with boundary, see [1]). 0. Notation. 1. By a manifold we shall always mean a smooth manifold; topologically it is a connected, paracompact space. For any

De Rham's theorem in a smooth topos

It has been persuasively argued (e.g. by Lawvere[8]) that the mathematical world picture needed to develop the physics of continuous bodies and fields should involve a cartesian closed category of smooth morphisms between smooth spaces. As far as the foundations of the calculus of variations are concerned, the need for such a category was recognized by K. T. Chen(cf. [2]).

A Remark on Whitney's Proof of de Rham's Theorem

A Remark on Whitney's Proof of de Rham's Theorem

Cup-product e prodotto esterno

Alternating cup-product for alternating singular cochains and cohomology classes, over a field of characteristic zero, is considered. Differently from ordinary cup-product, the alternating one is skew-commutative for cochains also, and exactly parallel to exterior product for differential forms. A direct proof of de Rham's third theorem follows, with use of a convenient simplicial (but not barycentric) subdivision.

Differential Forms and Maxwell's Field: An Application of Harmonic Integrals

Maxwell's equations are derived under the assumption of 4-dimensionality of Euclidean space from a somewhat different definition of time from that considered by Einstein, using differential forms and de Rham's theorem in the theory of harmonic integrals. It is shown that the continuity equation of the current density is an elementary consequence of the Jordan-Brouwer theorem of topology under the requirement of integrability of the field. Matter appears as a singular point of the field, and introduces various kinds of ``currents,'' in the sense of de Rham and Kodaira, according to the topological character of the domain of integration of the field. These ``currents'' describe characters of the material, and are represented not by ordinary functions, but by generalized functions in the Schwartz sense. Examples of these ``currents,'' such as electric and magnetic polarizations and the supercurrent, are given, and the origin of the fundamental difficulties with dimensions in the usual theory of electromagnetism is attributed to this fact.

On de Rham's theorem

THE CLASSICAL theorem of de Rham states, briefly, that the (real singular) cohomology of a (compact, smooth) manifold is naturally isomorphic to the cohomology derived from differential forms. There are many proofs of this in the literature, notably that by de Rham himself [4], that by Weil [6] (“pre-sheaf theory”), that by Schwartz [5] (“axiomatic”), and that by sheaf theory (Leray, e.g., in Godement’s book [2]). The purpose of the following lines is to give a different proof, using the handle decomposition of a manifold.

Acyclic Models and De Rham's Theorem

Acyclic Models and De Rham's Theorem

Acyclic models and de Rham’s theorem

The concept of acyclic models, developed by Eilenberg and MacLane [2], appears to be one of the most convenient tools of algebraic topology. It is the purpose of this paper to describe a generalization of the theory of Eilenberg and MacLane, and to apply the theory to proving de Rham's theorem. The generalized theory has other applications; in particular the proof of the spectral sequence theorem due to Gugenheim and Moore [4] could be slightly simplified by the results of this paper. The reader is assumed to know the theory of functors and categories; the general reference for this topic will be Eilenberg and Steenrod [3, Chapter IV, pp. 108-113]. The first two chapters are devoted to establishing the existence or uniqueness of chain and cochain maps; the third chapter is devoted to cup products. In the fourth chapter the singular Cr cubical cohomology theory is defined, and it is shown that for paracompact manifolds the result is independent of r. In the last chapter the de Rham cohomology is defined and shown to be isomorphic to the singular cohomology (de Rham's theorem) by means of the Stokes' map f* (integration over chains); it is also shown that the Stokes' map preserves products, i.e., that f*(ao3) = (f*a)U(f*). The relation of acyclic models to products will be treated at length in another paper.

De Rham's Theorem for Arbitrary Spaces

differentiable manifolds which can be defined in ter'ms of forms. The difficulty has been simply this: the Eilenberg-Steenrod method requires that cohomology groups be defined for all triangulable spaces, and not every triangulable space is a differentiable manifold. How to define forms on more general spaces than manifolds? Taking our clue from a fundamental property of differentiable manifolds, we make use of imbeddings in Euclidean spaces. The definition of the appropriate groups of forms is given in Section 1, together with the quite simple proofs of the seven cohomology axioms of Eilenberg-Steenrod. The groups can be defined for arbitrary compact subsets of Euclidean space; i. e., we get a cohomology theory for arbitrary compact finite dimensional spaces. In Section 2, we obtain the relationship

Differential Forms in Manifolds with Boundary

The theory of harmonic forms in Riemannian manifolds may be regarded as a generalization of potential theory. It is therefore natural that the boundary value problems of this theory which generalize the classical Dirichlet and Neumann problems should play an important role in the theory. Such theorems were conjectured by Tucker (4a). The present paper is concerned with some theorems regarding closed forms, which are necessary for the proof of the boundary value existence theorems for harmonic fields in a Riemannian manifold with boundary. In an accompanying paper, by D. C. Spencer and the author, the results of this paper will be used in the proofs of these theorems. We consider closed forms in a finite regular manifold with boundary, and prove results which generalize two theorems of G. de Rham (3) for a closed manifold. To extend the technique of de Rham, we make use of the relative, or modular, homology theory introduced by Lefschetz (2). The main results of the paper are contained in Theorems 2 and 6. Theorems 1, 3, and 5 are analogues of de Rham's first theorem, and Theorems 2, 4 and 6 of his second theorem. Theorems 3 to 6 inclusive were conjectured in substantially their present form by Tucker (4b).