The Biot -Savart operator and electrodynamics on bounded subdomains of the three -sphere
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, the set of points in R4 that are one unit away from the origin. By doing so, we establish a geometric setting for electrodynamics in positive curvature. When applied to a vector field, the Biot-Savart operator behaves like a magnetic field; we display suitable electric fields so that Maxwell's equations hold. Specifically, the Biot-Savart operator applied to a “current” V is a right inverse to curl; thus BS is important in the study of curl eigenvalue energy-minimization problems in geometry and physics. We show that the Biot-Savart operator is self-adjoint and bounded. The helicity of a vector field, a measure of the coiling of its flow, is expressed as an inner product of BS(V) with V. We find upper bounds for helicity on the three-sphere; our bounds are not sharp but we produce explicit examples within an order of magnitude. In all instances, the formulas we give are geometrically meaningful: they are preserved by orientation-preserving isometries of the three-sphere. Applications of the Biot-Savart operator include plasma physics, geometric knot theory, solar physics, and DNA replication.
Parsley, Robert Jason, "The Biot -Savart operator and electrodynamics on bounded subdomains of the three -sphere" (2004). Dissertations available from ProQuest. AAI3125886.