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

$p$-brane Galilean and Carrollian Geometries and Gravities