Briefly, one usually formulates the theory of gravity in terms of a a 4d spacetime with curvature but you can also formulate it as a theory of curved 3d shapes if you allow the lagrangian to carry more structure. This is often performed in GR, in fact, by decomposing the metric into a "spatial" and "temporal" part but shape dynamics kind of runs with this idea in an attempt to formulate a totally relational version of the theory of gravity.
Shape Dynamics apparently produces a reasonable theory of gravity which agrees with GR in many situations but forbids, I believe, closed timelike curves, and may be more amenable to quantization since it re-separates space and time.
Anyway, it all seems very beyond me, maybe even if I had the time, which I do not.