Marcos Salvai
Universidad Nacional de Córdoba, Argentina
Publications:
Emmanuele D., Salvai M., Vittone F.
Möbius Fluid Dynamics on the Unitary Groups
2022, vol. 27, no. 3, pp. 333351
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 biinvariant Riemannian
metric and has initially a homogeneous distribution of mass. By the least
action principle, forcefree 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.
