  We study steady-state magnetic fields in the geometric setting of positive
curvature on subdomains of the three-dimensional sphere. By generalizing the
Biot-Savart law to an integral operator BS acting on all vector fields, we show
that electrodynamics in such a setting behaves rather similarly to Euclidean
electrodynamics. For instance, for current J and magnetic field BS(J), we show
that Maxwell's equations naturally hold. In all instances, the formulas we give
are geometrically meaningful: they are preserved by orientation-preserving
isometries of the three-sphere.
  This article describes several properties of BS: we show it is self-adjoint,
bounded, and extends to a compact operator on a Hilbert space. For vector
fields that act like currents, we prove the curl operator is a left inverse to
BS; thus the Biot-Savart operator is important in the study of curl
eigenvalues, with applications to energy-minimization problems in geometry and
physics. We conclude with two examples, which indicate our bounds are typically
within an order of magnitude of being sharp.
