Herman Gluck

Vector Calculus and the Topology of Domains in 3-Space

Suppose you have a vector field defined on a bounded domain in 3-space. How can you tell whether your vector field is the gradient of some function? Or the curl of another vector field? Can you find a nonzero field on your domain that is divergence-free, curlfree, and tangent to the boundary? How about a nonzero field that is divergence-free, curl-free, and orthogonal to the boundary? To answer these questions, you need to understand the relationship between the calculus of vector fields and the topology of their domains of definition. The Hodge Decomposition Theorem provides the key by decomposing the space of vector fields on the domain into five mutually orthogonal subspaces that are topologically and analytically meaningful. This decomposition is useful not only in mathematics, but also in fluid dynamics, electrodynamics, and plasma physics. Furthermore, carrying out the proof provides a pleasant introduction to homology and cohomology theory in a familiar setting, and a chance to see both the general Hodge theorem and the de Rham isomorphism theorem in action. Our goal is to give an elementary exposition of these ideas. We provide three sections of background information early in the paper: one on solutions of the Laplace and Poisson equations with Dirichlet and Neumann boundary conditions, one on the Biot-Savart law from electrodynamics, and one on the topology of compact domains in 3-space. Near the end, we see how everything we have learned provides answers to the four questions that we have posed. We close with a brief survey of the historical threads that led to the Hodge Decomposition Theorem, and a guide to the literature.

The Biot–Savart operator for application to knot theory, fluid dynamics, and plasma physics

The writhing number of a curve in 3-space is the standard measure of the extent to which the curve wraps and coils around itself; it has proved its importance for molecular biologists in the study of knotted DNA and of the enzymes which affect it. The helicity of a vector field defined on a domain in 3-space is the standard measure of the extent to which the field lines wrap and coil around one another; it plays important roles in fluid dynamics and plasma physics. The Biot–Savart operator associates with each current distribution on a given domain the restriction of its magnetic field to that domain. When the domain is simply connected, the divergence-free fields which are tangent to the boundary and which minimize energy for given helicity provide models for stable force-free magnetic fields in space and laboratory plasmas; these fields appear mathematically as the extreme eigenfields for an appropriate modification of the Biot–Savart operator. Information about these fields can be converted into bounds on the writhing number of a given piece of DNA. The purpose of this paper is to reveal new properties of the Biot–Savart operator which are useful in these applications.

The spectrum of the curl operator on spherically symmetric domains

This paper presents a mathematically complete derivation of the minimum-energy divergence-free vector fields of fixed helicity, defined on and tangent to the boundary of solid balls and spherical shells. These fields satisfy the equation ∇×V=λV, where λ is the eigenvalue of curl having smallest nonzero absolute value among such fields. It is shown that on the ball the energy minimizers are the axially symmetric spheromak fields found by Woltjer and Chandrasekhar–Kendall, and on spherical shells they are spheromak-like fields. The geometry and topology of these minimum-energy fields, as well as of some higher-energy eigenfields, are illustrated.

Embedding and knotting of positive curvature surfaces in 3-space

Abstract In 3-space, compact orientable surfaces with nonempty boundary and positive curvature play the role of Seifert surfaces in a curvature-sensitive version of knot theory. The following result states that the isotopy classes of such surfaces are in a one-to-one correspondence with the isotopy classes of ordinary surfaces which have no constraint on their curvature.

Cohomology of harmonic forms on Riemannian manifolds with boundary

Abstract On a smooth compact manifold M, the cohomology of the complex of differential forms is isomorphic to the ordinary cohomology by the classical theorem of de Rham. When M has a Riemannian metric g, the harmonic forms constitute a subcomplex of the de Rham complex because the Laplacian commutes with exterior differentiation. When (M, g) has no boundary, all of its harmonic forms are closed, and hence the cohomology of this subcomplex is isomorphic to the ordinary cohomology by the classical theorem of Hodge. But when the boundary of (M, g) is non-empty, it is possible for a p-form to be harmonic without being closed, and some of these, which are exact, although not the exterior derivatives of harmonic p – 1-forms, represent an “echo” of the ordinary p – 1-dimensional cohomology within the p-dimensional harmonic cohomology.

Isoperimetric problems for the helicity of vector fields and the Biot–Savart and curl operators

The helicity of a smooth vector field defined on a domain in three-space is the standard measure of the extent to which the field lines wrap and coil around one another. It plays important roles in fluid mechanics, magnetohydrodynamics, and plasma physics. The isoperimetric problem in this setting is to maximize helicity among all divergence-free vector fields of given energy, defined on and tangent to the boundary of all domains of given volume in three-space. The Biot–Savart operator starts with a divergence-free vector field defined on and tangent to the boundary of a domain in three-space, regards it as a distribution of electric current, and computes its magnetic field. Restricting the magnetic field to the given domain, we modify it by subtracting a gradient vector field so as to keep it divergence-free while making it tangent to the boundary of the domain. The resulting operator, when extended to the L2 completion of this family of vector fields, is compact and self-adjoint, and thus has a largest ...

The embedding of two-spheres in the four-sphere

We consider the question of reducing the homeomorphism problem for pairs of topological spaces to the homeomorphism problem for single spaces. Of particular interest is the strength of the invariant X-A of the pair (X, A). In the first chapter we discuss the problem in its general setting. In the second chapter we consider the embedding of n-spheres in the n+2 sphere, and in particular the case n=2. Instead of the complement Sn+2-Sn, we use the exterior of Sn in Sn+2 (i.e., the complement of an open regular neighborhood of Sn). We prove that there are at most two nonequivalent locally flat embeddings of S2 in S4 with homeomorphic exteriors. In certain classical cases, the exterior is a complete invariant for the embedding. In the third chapter we consider the structure of embeddings in more detail. The main result is that any locally flat orientable surface in four-space is the boundary of an orientable three-manifold in four-space. In the fourth and final chapter, we consider the embedding of n-spheres in the n+2 sphere for values of n other than 2. The situation for n> 2 resembles that for n =2, while the case n = 1 stands apart from the rest.

Manifolds with preassigned curvature—a survey

In this paper I discuss two problems of Riemannian geometry in the large concerning the existence of manifolds with preassigned curvature. The Minkowski problem and its generalization asks in Euclidean space for a closed convex hypersurface whose curvature has been given in advance. The converse to the Gauss-Bonnet theorem asks for the existence, on a two-dimensional manifold, of a Riemannian metric with prescribed Gaussian curvature. The questions have a meeting point: the search for two-spheres in three-space with given strictly positive curvature. While the first problem goes back to the work of Minkowski [32] in 1897, the second is of more recent vintage: it was posed explicitly by Warner in the early 1960s. Both have been solved in the last few years, and in this survey I try to give an overview and some of the details. The paper is organized into the following sections: 1. The Minkowski problem 2. The generalized Minkowski problem 3. Converse to the Gauss-Bonnet theorem for smooth manifolds 4. Converse to the Gauss-Bonnet theorem for PL manifolds 5. Realization in three-space

Electrodynamics and the Gauss linking integral on the 3-sphere and in hyperbolic 3-space

We introduce here explicit integral formulas for linking, twisting, writhing and helicity on the 3-sphere and in hyperbolic 3-space. These formulas, like their prototypes in Euclidean 3-space, are geometric rather than just topological, in the sense that their integrands are invariant under orientation-preserving isometries of the ambient space. They are obtained by developing and then applying a steady-state version of classical electrodynamics in these two spaces, including an explicit Biot-Savart formula for the magnetic field and a corresponding Amperes law contained in Maxwells equations. The Biot-Savart formula leads, in turn, to upper bounds for the helicity of vector fields and lower bounds for the first eigenvalue of the curl operator on subdomains of the 3-sphere and hyperbolic 3-space. We give only a hint of the proofs.

The inaudible geometry of nilmanifolds.

SummaryWe show that isospectral deformations of compact Riemannian two-step nilmanifolds can be systematically detected by simple changes in the behavior of their geodesics, in spite of the fact that the length spectrum (which measures the lengths of all closed geodesics) remains constant.