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Marcos Salvai

Universidad Nacional de Córdoba, Argentina


Emmanuele D., Salvai M., Vittone F.
Möbius Fluid Dynamics on the Unitary Groups
2022, vol. 27, no. 3, pp.  333-351
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.
Keywords: force-free motion, kinetic energy metric, nonrigid dynamics, unitary group, split unitary group, M¨obius action, maximal isotropic subspace, inviscid Burgers equation
Citation: Emmanuele D., Salvai M., Vittone F.,  Möbius Fluid Dynamics on the Unitary Groups, Regular and Chaotic Dynamics, 2022, vol. 27, no. 3, pp. 333-351

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