As advocated by Élie Cartan long ago, the geometrisation of gravity induced by the equivalence principle is by no means restricted to general relativity. In this light, Einstein’s and Newton’s theories of gravity both admit geometrical formulations which are diffeomorphism invariant. We will review Newton-Cartan geometry and its recent torsional extension which found some applications in condensed matter physics. The "metric" structure of nonrelativistic spacetimes consists of a one-form (an absolute clock) whose kernel is endowed with a positive-definite metric (absolute rulers). Connections compatible with degenerate metric structures are known to possess peculiar features: on the one hand, the compatibility conditions involve restrictions on the torsion; on the other hand, torsionfree compatible connections are not unique. The complete classification of Galilean and Carrollian connections with torsion and their relativistic lift will be presented.

NOTES

Note unusual time