$p$-brane Galilean and Carrollian Geometries and Gravities

We study $D$-dimensional $p$-brane Galilean geometries via the intrinsic torsion of their adapted connections. These non-Lorentzian geometries are examples of $G$-structures whose characteristic tensors consist of two degenerate ``metrics'' of ranks $(p+1)$ and $(D-p-1)$. We carry out the analysis in two different ways. In one way, inspired by Cartan geometry, we analyse in detail the space of intrinsic torsions (technically, the cokernel of a Spencer differential) as a representation of $G$, exhibiting for generic $(p,D)$ five classes of such geometries, which we then proceed to interpret geometrically. We show how to re-interpret this classification in terms of ($D-p-2$)-brane Carrollian geometries. The same result is recovered by methods inspired by similar results in the physics literature: namely by studying how far an adapted connection can be determined by the characteristic tensors and by studying which components of the torsion tensor do not depend on the connection. As an application, we derive a gravity theory with underlying $p$-brane Galilean geometry as a non-relativistic limit of Einstein--Hilbert gravity and discuss how it gives a gravitational realisation of some of the intrinsic torsion constraints found in this paper. Our results also have implications for gravity theories with an underlying ($D-p-2$)-brane Carrollian geometry.