On the intrinsic torsion of spacetime structures

We briefly review the notion of the intrinsic torsion of a $G$-structure and then go on to classify the intrinsic torsion of the $G$-structures associated with spacetimes: namely, galilean (or Newton-Cartan), carrollian, aristotelian and bargmannian. In the case of galilean structures, the intrinsic torsion classification agrees with the well-known classification into torsionless, twistless torsional and torsional Newton-Cartan geometries. In the case of carrollian structures, we find that intrinsic torsion allows us to classify Carroll manifolds into four classes, depending on the action of the Carroll vector field on the spatial metric, or equivalently in terms of the nature of the null hypersurfaces of a lorentzian manifold into which a carrollian geometry may embed. By a small refinement of the results for galilean and carrollian structures, we show that there are sixteen classes of aristotelian structures, which we characterise geometrically. Finally, the bulk of the paper is devoted to the case of bargmannian structures, where we find twenty-seven classes which we also characterise geometrically while simultaneously relating some of them to the galilean and carrollian structures. This paper is dedicated to Dmitri Vladimirovich Alekseevsky on his 80th birthday.