Mobius fluid dynamics on the unitary groups]{M\"{o}bius fluid dynamics on the unitary groups

TL;DR

This paper investigates the geometric and dynamical properties of nonrigid motions induced by the split unitary groups acting on classical Lie groups, revealing their geodesic structure, symmetries, and relations to conformal and projective sphere motions.

Contribution

It characterizes the geometry of the kinetic energy metric on these groups, identifies symmetries and geodesic submanifolds, and links the dynamics to Burgers equations with M"{o}bius constraints.

Findings

Kinetic energy metric on G is not complete or invariant.

Identified symmetries and totally geodesic submanifolds of G.

Established conditions when geodesics of rigid motions are geodesics of G.

Abstract

We study the nonrigid dynamics induced by the standard birational actions of the split unitary groups $G=O_{o}\left( n,n\right) $, $SU\left( n,n\right) $ and $Sp\left( n,n\right) $ 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 least action 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 M\"{o}bius 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.