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

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

2004-06-14

arxiv.org/abs/math/0406276v1

Mathematics

Geometric Topology

Scientific paper

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 Ampere's law contained in Maxwell's 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.

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