Generalized Galilean Geometries

Motivated by non-relativistic string theory, we give a classification of D-dimensional generalized Galilean geometries. They are an extension of the Galilean geometry in the sense that the two non-degenerate metrics of Galilean geometry (one to measure time intervals and another one to measure spatial distances) are replaced by two non-degenerate metrics of rank p+ 1 and rank D- p- 1, respectively, with p= 0, 1, ⋯, D- 1. To classify these generalized geometries an important role is played by the so-called intrinsic torsion tensor indicating that this particular torsion is independent of the spin-connection. We show that there is a finite way of setting some of these intrinsic torsion tensors equal to zero and that this leads to a classification of the generalized Galilean geometries. Moreover, we show how some (but not all) of the generalized Galilean geometries that we find can be obtained by taking a special limit of general relativity.