  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.
