Exterior Differentiation Research Articles

Fermions and bosons in a unified framework. III. Mathematical structures and physical questions

By examining the implications of the usual physical assumptions for gauge theories we are led to a number of interesting structures which can be formulated in terms of moving frames. The first problem we consider is Faddeev's fiber bundle procedure for removal of gauge degeneracies from quantum fields. We show how the theory of moving frames can be related to the theory of fiber bundles. In the fiber bundle extra displacements and extra components of the gauge fields are included. These precisely remove the gauge degeneracies. In order to formulate this problem in the language of fiber bundles, we begin with a section detailing the algebra of forms in four dimensions and their associated operations (exterior differentiation, Hodge duality and the natural Hilbert product of forms). We then show how to form the action for the SO(3) fiber bundle. This opens the general question (implicitly answered by this example) of how to quantize in a fiber bundle. Next we examine the question of choice of local symmetry structure. First we consider a space whose local rotations do not form a group. By generalizing the structure equations for manifolds, we can study such a space. Second, in general relativity it is assumed that at any point one can choose an orthonormal basis and a vanishing connection. This corresponds to giving the zeroth, first, and second coefficient of a Taylor's expansion of the coordinate system. But manifolds flatter than those of general relativity do exist for which the structure is carried in the higher coefficients. We show how to give an action principle for them. Third, algebras more general than Lie algebras are possible local symmetries, e.g., superalgebras, the octonian algebra, and others violating the Jacobi relation. An action is given for a one-parameter family of quasi-Lie algebras which reduces to that of u(1) + su(2) in the zero limit of the parameter. Fourth, simple examples of actions for gauged superalgebras are given which do not include the full complexity of general relativity. Fifth, an easy derivation of the supergravity action is made; then generalized to a supergravity Weinberg-Salam-type model.

The exterior algebra for Wiemann manifolds

We discuss the theory of infinite-dimensional manifolds from the point of view of establishing a widely applicable framework for generalization of the finite-dimensional Hodge theory. The principal result is the development of an exterior algebra based on a weakened definition of differentiation, so that “ C ∞” partitions of unity are available for paracompact manifolds modelled on arbitrary real separable Banach spaces. We prove a Poincaré lemma for our new notion of exterior differentiation, and go on to discuss the relationship of the exterior derivative with current research efforts toward the definition of an infinite-dimensional Laplace-Beltrami operator.

Exterior differentiation in Lie algebras

Exterior differentiation in Lie algebras

Local Invariants of an Embedded Riemannian Manifold

Let M be a Riemannian manifold of dimension m which is isometrically embedded in a Riemannian manifold N of dimension me + r. Let G denote the metric on N and GM denote the metric on-M. In this paper, we will study the local invariants of such an embedding. The invariants will be p-form valued polynomials in the derivatives of the metric defined on M. We have studied the case r = 0 in [4] for pseudo-Riemannian manifolds M. The results of this paper can be generalized to the case in which both M and N are pseudo-Riemannian manifolds. We will be considering invariance relative to the following four groups: let v = v, x ,29 Set v, = SO(m) if M is oriented and 0(m) otherwise, ,2 = SO(r) if the normal bundle is oriented and 0(r) otherwise. Let P , denote the space of p-form valued >-invariant polynomials in the derivatives of the metric; we will give a precise definition of these spaces in the first section. Since exterior differentiation is natural, d induces a map d: 9P CP +J. If p < n, the kernel of d modulo the image of d is spanned by the characteristic classes. If p = m, every form is closed. We will give an integral condition which characterizes those invariants which are characteristic forms + dQ for some Q e gJ m-l In Corollary 1.3, we apply these theorems to settle a conjecture of I. M. Singer regarding the Euler class (private communication).

Space tensors in general relativity III: The structural equations

A self-consistent theory of spatial differential forms over a pair (M,Γ)is proposed. The operators d(spatial exterior differentiation), dT (temporal Lie derivative) andL (spatial Lie derivative) are defined, and their properties are discussed. These results are then applied to the study of the torsion and curvature tensor fields determined by an arbitrary spatial tensor analysis $$(\tilde \nabla ,\tilde \nabla T)$$ (M,Γ). The structural equations of $$(\tilde \nabla ,\tilde \nabla T)$$ and the corresponding spatial Bianchi identities are discussed. The special case $$(\tilde \nabla ,\tilde \nabla T) = (\tilde \nabla *,\tilde \nabla T*)$$ is examined in detail. The spatial resolution of the Riemann tensor of the manifold M is finally analysed; the resultingstructure of Eintein's equations over a pair (ν4,Γ)is established. An application to the study of the problem of motion in terms of co-moving atlases is proposed.

Removable singularities of solutions of linear partial differential equations

Suppose P(x, D) is a linear partial differential operator on an open set ~ contained in R ~ and that A is a closed subset of ~. Given a class ~(~) of distributions on ~, the set A is said to be removable for ~(~) if e ach /E ~(~), which satisfies P(x, D ) / = 0 in ~ A , also satisfies P(x, D)/= 0 in ~. The problem considered in this paper is the following. Given a class ~(~) of distributions on ~, what restriction on the size of A will ensure that A is removable for ~(~). We obtain results for Lroc (~) (p ~< ~ ) , C(~), and Lipa (~). The first result of this kind was the Riemann removable singularity theorem: if a function / is holomorphic in the punctured unit disk a n d / ( z ) = o ( H -1) as z approaches zero, then / is holomorphic in the whole disk. Bochner [1] generalized Riemann's result by considering the class ~(~) of functions f on ~ such that ](x)=o(d(x, A) -q) uniformly for x in compact subsets of ~, and giving a condition on the size of A which insures tha t A is removable for ~(~) (Theorem 2.5 below). Bochner's theorem is remarkable in that the condition on the size of A only depends on the order of the operator P(x, D). The theorem applies, therefore, to systems of differential operators, such as exterior differentiation in R n and ~ (the Cauchy-Riemann operator) in C n. The same can be said for the other results in this paper. The proof of Bochner's theorem provided the motivation for our results. I t is interesting to note tha t a very general result (Corollary 2.4) f o r / ~ (~) (due to Li t tman [7]) is an easy corollary of Bochner's work. Here the condition on the singular set A is expressed in terms of Minkowski content. In section 4 the case of Ll~oc(~) is studied again, and results in section 2 are improved by replacing Minkowski content with Hausdorff measure. In addition, the cases C(~)

Resolutions by hyperfunctions of sheaves of solutions of differential equations with constant coefficients

Recently R. HARVEY [-4] and G. BENGEL [1] proved the existence and regularity theorems for hyperfunction solutions of single differential equations with constant coefficients. We extend their results to systems of differential equations with constant coefficients. Since the sheaf of hyperfunctions is flabby, we have therefore a flabby resolution of the sheaf of hyperfunction solutions of the homogeneous equation. If the system is elliptic, it is shown that any hyperfunction solution is analytic. Thus we obtain a concrete flabby resolution of the sheaf of regular solutions in terms of hyperfunctions. We apply this resolution first to the Cauchy-Riemann system and to single elliptic operators, and give a proof of the vanishing cohomology theorem by MALGRANGE [-10] for coherent analytic sheaves and G.BENGEL'S results [1] on P-functionals. Then applying the resolution to the exterior differentiation, we obtain a kind of the Alexander-Pontrjagin duality theorem and hence a purely analytic proof of the Jordan-Brouwer theorem.

Remark on formal exterior differentiation

Remark on formal exterior differentiation

Symmetric Exterior Differentiation and Flat Forms

Let ω be a continuous differential r-form defined in a bounded domain R of Euclidean n-space, En, where n ≧ 1 and 0 ≦ r ≦ n — 1. ω is called a flat form in R, (3, p. 263), if there exists a constant N such that for every (r + 1)-simplex σ contained in R, where |σ| designates the (r + 1)-volume of σ. For n = 1 and ω a zero form, flatness is the same thing as the usual Lip 1 condition. As is well known, a necessary and sufficient condition that a continuous real-valued function of one variable f(x) satisfy a Lip 1 condition over an interval (a, b) is that its upper and lower symmetric derivatives be bounded in (a, b) (4, pp. 22, 327). We intend to prove the analogue of this result for r-forms.

On the complex form of the Poincaré lemma

dimension n, any differential form can be decomposed into a sum of forms of degree r, 0 < r < n, and the operator d of exterior differentiation (d2 = 0) is an operator of degree + 1. A differential form q satisfying do =0 is called closed, and a form q such that q =dfl is called exact. The Poincare Lemma states that a closed differential form of positive degree is locally exact; that is, if q is defined and satisfies dqb = 0 in an open UCM, and mo is any point of U, then there exists an open neighborhood V of mo, with VC U, and a form 41 defined in V such that ck = d,6 in V. This result is most conveniently proved by choosing a V differentiably contractible to the point mo, for which there can be constructed an operator k of degree -1 satisfying