  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.

High Energy Physics - Theory;Differential Geometry