  Simply-connected homogeneous spacetimes for kinematical and aristotelian Lie
algebras (with space isotropy) have recently been classified in all dimensions.
In this paper, we continue the study of these "maximally symmetric" spacetimes
by investigating their local geometry. For each such spacetime and relative to
exponential coordinates, we calculate the (infinitesimal) action of the
kinematical symmetries, paying particular attention to the action of the
boosts, showing in almost all cases that they act with generic non-compact
orbits. We also calculate the soldering form, the associated vielbein and any
invariant aristotelian, galilean or carrollian structures. The (conformal)
symmetries of the galilean and carrollian structures we determine are typically
infinite-dimensional and reminiscent of BMS Lie algebras. We also determine the
space of invariant affine connections on each homogeneous spacetime and work
out their torsion and curvature.
