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

Abstract

In this first of two papers, we develop a steady-state version of classical electrodynamics on the 3-sphere and in hyperbolic 3-space, including an explicit formula for the vector-valued Green’s operator, an explicit formula of Biot–Savart type for the magnetic field, and a corresponding Ampere’s law contained in Maxwell’s equations. We then use this to obtain explicit integral formulas for the linking number of two disjoint closed curves in these spaces. These formulas, like their prototypes in Euclidean 3-space, are geometric rather than just topological because their integrands are invariant under orientation-preserving isometries of the ambient space. In the second paper, we obtain integral formulas for twisting, writhing, and helicity and prove the theorem LINK=TWIST+WRITHE in the 3-sphere and in hyperbolic 3-space. We then use these results to derive upper bounds for the helicity of vector fields and lower bounds for the first eigenvalue of the curl operator on subdomains of these two spaces. An announcement of most of these results and a hint of their proofs can be found in e-print arXiv:math.GT/0406276, while an expanded version of this paper, containing many details which are here left to the reader, can be found at e-print arXiv:math.GT/0510388.