Reduced Singular Solutions of EPDiff Equations on Manifolds with Symmetry

The EPDiff equation for geodesic flow on the diffeomorphisms admits a remarkable ansatz for its singular solutions, called “diffeons.” Because this solution ansatz is a momentum map, the diffeons evolve according to canonical Hamiltonian equations. We examine diffeon solutions on Einstein spaces that are “mostly” symmetric, i.e., whose quotient by a subgroup of the isometry group is 1-dimensional. An example is the two-sphere, whose isometry group SO(3) contains S. In this situation, the canonical Hamiltonian dynamics for the diffeons reduces from (integral) partial differential equations to ordinary differential equations in time. We analyse the basic diffeon solutions of these canonical equations for the sphere and for several other 2-dimensional examples. Explicit calculations are provided for future numerical implementation. From consideration of these 2-dimensional spaces, we then begin developing the theory for a general manifold possessing a metric equivalent to the warped product of the line with the bi-invariant metric of a Lie group.