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Properties of the exterior algebra of a vector space are used to investigate the curvature operator of a Riemannian manifold. Induced inner products and linear maps are used to establish results about the Euler characteristic of a compact manifold. An open problem about the decomposition of operators on A 2 V is discussed. This problem arises in the study of the codimension needed for isometric embeddings. A new algebraic consequence of the first Bianchi identities is established.

The variation of Hodge structure defined by the natural family of hypersurfaces of degreed and dimensionn is maximal if the cohomology has Hodge level >1. There is a small list of hypersurfaces of “level one” which give non-maximal variations: plane curves of degreed≧5, cubics of dimension 3 and 5, and quartic threefolds.

To each finite-dimensional vector space E can canonically associate the space ΛE * ⊗ ΛE. This space has a natural inner product (bilinear, symmetric and non-degenerate) and two (algebraic) product structures. Identities relating these products should be important for understanding the structure of these algebras. Indeed such identities have been used, both to develop Mixed Exterior Algebra and to apply it. In the present paper, we obtain a new identity of this type and show that the identities mentioned above are consequences of it. In this sense, it may be regarded as basic to the subject.

By applying the Hodge decomposition theorem to the local number and current densities of the Bose liquid, we have been able to derive the interaction Hamiltonian for the two-dimensional vortex gas. Our approach relies upon looking at the fields residing in the kernel of the Laplace operator.

This paper represents an detailed version of our paper [12] presented at the Hawaii Conference on the Geometry of the Laplace Operator, March 1979. In this paper we present the solution to the following problem posed by Singer in [8], w 4, concerning elliptical operators on PL-maifolds: "If M is a PL-manifold, the L-polynomials are still well defined (Thom [13]) from which one can still define the rational Pontrjagin classes. The Hirzebruch signature theorem still holds. Is there an associated elliptic operator (as in the smooth case) whose index is the signature? On what spaces does it operate? Does it have a symbol and where does the symbol lie?"

(1983). Hodge theory on domains with conic singularities. Communications in Partial Differential Equations: Vol. 8, No. 1, pp. 65-88.

The paper deals mainly with combinatorial structures; in some cases we need refinements of combinatorial structures. Riemannian metrics are defined on any combinatorial manifold M. The existence of distance functions and of Riemannian metrics with “constant volume density” implies smoothing. A geometric realization of PL ( m ) /O ( m ) {\text {PL}}\left ( m \right ){\text {/O}}\left ( m \right ) is given in terms of Riemannian metrics. A graded differential complex Ω ∗ ( M ) {\Omega ^ {\ast } }( M ) is constructed: it appears as a subcomplex of Sullivan’s complex of piecewise differentiable forms. In the complex Ω ∗ ( M ) {\Omega ^{\ast }}( M ) the operators d d , ∗ \ast , δ \delta , Δ \Delta are defined. A Rellich chain of Sobolev spaces is presented. We obtain a Hodge-type decomposition theorem, and the Hodge homomorphism is defined and studied. We study also the combinatorial analogue of the signature operator.

We define a notion of a stable system of Hodge bundles. A stable system of Hodge bundles has a Hermitian-Yang-Mills metric and, if certain Chern classes vanish, this gives a complex variation of Hodge structure. We use these ideas to obtain a criterion for a variety to be uniformized by a bounded symmetric domain.

In four dimensions, two metrics that are conformally related define the same Hodge dual operator on the space of two‐forms. The converse, namely, that two metrics that have the same Hodge dual are conformally related, is established. This is true for metrics of arbitrary (nondegenerate) signature. For Euclidean signature a stronger result, namely, that the conformal class of the metric is completely determined by choosing a dual operator on two‐forms satisfying certain conditions, is proved.

We study some explicit relations between the canonical line bundle and the Hodge bundle over moduli spaces for low genus. This leads to a natural measure on the moduli space of every genus which is related to the Siegel symplectic metric on Siegel upper half-space as well as to the Hodge metric on the Hodge bundle.

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 Poincare 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.

It is the purpose of this paper to find necessary and sufficient conditions for an exterior p-form zu to factor into a product ZL’ = t A y1 h yz A ... A yTc of k l-forms, and a (p K)-f orm; and to establish a maximum fork (Theorem 1). This is done by introducing a certain invariant for w called “length” which is subsequently proved to be equal to the “rank” of the dual (E p)vector (Theorem 2). An interesting application is that an (212 I)-form on an (2% + l)dimensional vector-space always f;ictors into a product of a l-form and a (272 2)-form (Corollary 2.1). One also obtains a quick proof for the standard theorem that an exterior p-form w is fully-decomposable (i.e=, a product of l-forms) if and only if it is of minimal rankp, and that p + 2 < Y(ZL’) < n when ‘W is not fully decomposable (Corollary 2.2).

A simple formula is derived for the infinitesimal change of the Hodge dual of a k-form, induced by a deformation of the scalar product in the underlying vector space. By considering deformations due to a flow generated by a vector field on a differential manifold, one obtains an expression for the commutator of the Hodge dual with the Lie derivation with respect to the vector field, acting on differential forms. This formula is useful in proving theorems on optical solutions of Maxwell's and Yang-Mills equations. The optical geometry underlying such solutions is defined as a restriction of the bundle of linear frames of a 4-dimensional manifold to a 9-dimensional optical group. This geometry provides a natural framework for the study of shearfree, optical and geodesic congruences and of the associated fields.

Let E be a vector space of dimension n ∊ N over a field Γ, of characteristic 0. Choose E*, dual to E, and formthe space of exterior powers of E, as well as ΛE*. Finally, let

We specialize the gauge-fixing procedure for the Witten action of the open bosonic string, given in a preceding paper, choosing the Siegel gauge. We find that the BRST-invariant gauge-fixed action is the gauge invariant one with ghost number unrestricted plus a gauge-fixing term. The BRST invariance of the measure in the functional integral is briefly discussed. As a technical tool the Hodge dual of a string functional is defined.

Any theory of choice under uncertainty in which choice depends on locally linear evaluations of univariate probability density functions has precisely two testable implications. These are 1. (a) d ũ = Ω 2. (b) ★d★ ũ = φ. Here φ, ũ , and Ω are differential forms, φ a zero-form, ũ a one-form, and Ω a two-form. d is the exterior derivative operator on forms, and ★ is the Hodge star operator. In this letter we explain in a non-technical way the meaning and import of equations (a) and (b). A fuller treatment will be found in Russell (1989).

We consider Maxwell's system with anisotropic coefficients, i.e., the electric permittivity and the magnetic permeability are assumed to be matrices with non-analytic entries. Under the additional assumption that one matrix is a scalar multiple of the other we prove unique continuation of the system across non-characteristic surfaces. The proof relies on differential geometry as well as on Carleman estimates.

An L-2 theory of differential forms is proposed for the Banach manifold of continuous paths on a Riemannian manifold M furnished with its Brownian motion measure. Differentiation must be restricted to certain Hilbert space directions, the H-tangent vectors. To obtain a closed exterior differential operator the relevant spaces of differential forms, the H-forms, are perturbed by the curvature of M. A Hodge decomposition is given for L-2 H-one-forms, and the structure of H-two-forms is described. The dual operator d* is analysed in terms of a natural connection on the H-tangent spaces. Malliavin calculus is a basic tool. (c) 2007 Elsevier Inc. All rights reserved.

The usual proof that the Hodge star operator on the Grassmann algebra is independent of the orthonormal basis used to define it requires a decomposition of the operator into a product of three maps, each of which is independent of the basis. The present note contains a very short proof of the result which depends on simple properties of the multiplication and induced inner product in the Grassmann algebra.

with boundary condition lim|x|_>oo |$|(z) = 1. Here, the notation follows [1]. That is, FA is the curvature of A, DA is the exterior covariant derivative on f\T* E and [•,•] is the natural, graded bracket on /\T* E : If w, 77 are, respectively, unvalued p, q forms, then [w, 77] = w A r\ — (—l)rj A LJ. The * in (1) is the Hodge star on f\T* from the Euclidean metric on T*. The norm | • | on T* 0 E ;i8 that induced from the Euclidean metric on T* and the Killing metric on su(2). Equation (1) is the variational equation of an action functional

An introductory review is presented of differential forms and their use in physics, particularly with regard to gauge theories. The subjects covered are: 1. Vector fields 2. Differential forms 3. The wedge product 4. The exterior derivative 5. Integration 6. Vector valued differential forms 7. Frames 8. Metrics on a vector space 9. Metrics on an open subset of Rn 10. The Hodge star 11. The coderivative 12. The Laplace operator 13. Summary of the mathematical part 14. Maxwell’s equations 15. Gauge theories and the Yang-Mills action 16. The equivalence principle 17. The Einstein-Cartan equations 18. A farewell to ω

Our discussion of differential forms is spread over the next four chapters. The fundamentals of the algebraic theory and theory of integration of differential forms form the substance of Chapter 6. This material is a necessary expansion of the introductory material in James (1954). Chapter 7 is concerned with explicit derivation of differential forms for invariant measures on various manifolds. Although somewhat premature, in case the manifold is a locally compact metric matrix group, the basic idea has already been discussed in Chapter 3, Section 3.6. We discuss several manifolds, the Stiefel and Grassman manifolds, which are not groups, in Chapter 7. These examples require some long detailed calculations and this material has been segregated into a separate chapter. The fact that differential forms are so useful in the derivation of density functions is due to the fact that the manifold of n × h matrices naturally decomposes into a topological product of manifolds on which groups of transformations act. The theory of Sections 3.3 and 3.4 suggests that the invariant measures should factor. Since Lebesgue measure is absolutely continuous relative to the invariant measures, Lebesgue measure also factors. Chapter 8 discusses the well-known methods of decomposing a n × h matrix. Chapter 9 factors Lebesgue measure over several of these products of manifolds thereby obtaining density functions of several multivariate statistics.

Introduction Mixed Hodge structures of a stable curve The topology of the versal deformation of a stable curve Limit Hodge structure of the versal deformation The Voronoi invariant Formulation of the main theorem The period matrix of 1-motive First approximation to the nilpotent orbit Second approximation to the nilpotent orbit References.

The geometry of Yang-Mills fields is placed in the general frame of the Hamilton-Cartan formalism of the Calculus of Variations. The pre-symplectic structure of the space of solutions of the linearization of field equations at one of their solutions is studied. Using results of the Hodge theory for harmonic forms, the radical of the pre-symplectic metric of a Yang-Mills field is determined and conclusions are drawn about the corresponding manifolds of moduli with respect to the gauge group. The procedure to be followed in order to generalize the method to "minimal interactions" is illustrated with an elementary example, and finally some possible ways for further generalizations are pointed out.

Comparing the dimensions of the two spaces S p (E) and S n–p (E) we see that these spaces are isomorphic.

This paper concerns the characterization of tangential traces for the space H(curl, Ω), when Ω is a Lipschitz polyhedron in ℝ3 under general topology assumptions. Suitable spaces of tangential vector fields are introduced and characterized. Hodge decompositions are provided and they involve the first cohomology space of the boundary Γ. Such a space is characterized and a basis is furnished.

Some properties of the operator algebra generated by Hodge’s star and the exterior derivative are established and in particular it is shown that viewed as an algebra over its center, this algebra is four‐dimensional.