An intrinsic Biot–Savart Law and Self-Induction on C1 immersions

We derive an intrinsic Biot—Savart law determined solely by the geometry and topology of a vortex filament, with no reference to the ambient space S 3 . The resulting self-induced velocity potential belongs to L q ( T S 3 ) for 1 < q < 3 2 and satisfies an L 2 reproducing identity, implying strong spatial concentration around the filament. The construction applies to all C 1 immersions of [0,1] and S 1 into S 3 , providing a unified, geometry-based formulation of self-induced motion for singular vorticity distributions. As an application we write the Euler equations of hydrodynamics on the manifold of C 1 immersions.