Intrinsic equations for the kinematics of a classical vortex string in higher dimensions

We present here the natural algebraic-geometrical generalization of the Da Rios Betchov intrinsic equations governing curvature and torsion of an isolated vortex string moving in an unbounded, perfect fluid flow. The filament is embedded in a manifold that can be assumed to be homeomorphic to an odd-dimensional Euclidean space, and whose connection we do not assume to be torsion-free. We suggest how to account for fluid compressibility in the ambient space by its geometrization, and we discuss some special cases of physical interest such as the torsion-free affine connection case and the Riemannian connection case. Finally, we point out the role our results might have in the context of soliton studies.