Gauge theory of Riemann ellipsoids

Abstract

The classical theory of Riemann ellipsoids is formulated naturally as a gauge theory based on a principal G-bundle ${\cal P}$. The structure group G=SO(3) is the vorticity group, and the bundle ${\cal P}=GL_+(3, R})$ is the connected component of the general linear group. The base manifold is the space of positive-definite real $3\times 3$ symmetric matrices, identified geometrically with the space of inertia ellipsoids. The bundle ${\cal P}$ is also a Riemannian manifold whose metric is determined by the kinetic energy. Nonholonomic constraints determine connections on the bundle. In particular, the trivial connection corresponds to rigid body motion, the natural Riemannian connection - to irrotational flow, and the invariant connection - to the falling cat. The curvature form determines the fluid's field tensor which is an analogue of the familiar Faraday tensor.