The Biot-savart Operator and Electrodynamics on Subdomains of the Three-sphere

ABSTRACT. We study the generalization of the Biot-Savart law from electrodynamics in the presence of curvature. We define the integral operator BS acting on all vector fields on subdomains of the three-dimensional sphere. By doing so, we establish a geometric setting for electrodynamics in positive curvature. When applied to a vector field, the BiotSavart operator behaves like a magnetic field; we demonstrate that Maxwell’s equations hold for this context. In particular, for vector fields that act like currents, the curl operator is a left inverse to BS; thus the Biot-Savart operator is important in the study of curl eigenvalue energy-minimization problems in geometry and physics. We show that the BiotSavart operator is self-adjoint and bounded. In all instances, the formulas we give are geometrically meaningful: they are preserved by orientation-preserving isometries of the three-sphere.