Möbius fluid dynamics on the unitary groups

Abstract

We study the nonrigid dynamics induced by the standard birational actions of the split unitary groups G=O_o(n,n), SU ( n,n) and Sp(n,n) on the compact classical Lie groups M = SO_n, U_n and Sp_n, respectively. More precisely, we study the geometry of G endowed with the kinetic energy metric associated with the action of G on M, assuming that M carries its canonical bi-invariant Riemannian metric and has initially a homogeneous distribution of mass. By the leastaction principle, force free motions (thought of as curves in G) correspond to geodesics of G. The geodesic equation may be understood as an inviscid Burgers equation with Möbius constraints. We prove that the kinetic energy metric on G is not complete and in particular not invariant, find symmetries and totally geodesic submanifolds of G and address the question under which conditions geodesics of rigid motions are geodesics of G. Besides, we study equivalences with the dynamics of conformal and projective motions of the sphere in low dimensions.