On Galilean isometries

We introduce three nested Lie algebras of infinitesimal ‘isometries ’ of a Galilei space-time structure which play the rôle of the algebra of Killing vector fields of a relativistic Lorentz spacetime. Non trivial extensions of these Lie algebras arise naturally from the consideration of Newton-Cartan-Bargmann automorphisms. Quite recently, Carter and Khalatnikov [CK] have pointed out that a geometric four-dimensional formulation of the non relativistic Landau theory of perfect superfluid dy-namics should involve not only Galilei covariance but also, more significantly as far as gravitational effects are concerned, covariance under a larger symmetry group which the