Kodaira Families and Newton-Cartan Structures with Torsion

We describe the induced geometry on several classes of Kodaira moduli spaces of rational curves in twistor spaces. By constructing connections and frames on the moduli spaces we build and review twistor theories pertaining to relativistic and non-relativistic geometries. Focussing on the cases of three- and five-dimensional moduli spaces we establish novel twistor theories of Newton-Cartan spacetimes. We generalise a canonical class of connections on Kodaira moduli spaces to encompass torsion and prove that in three dimensions deformations of the twistor space's holomorphic structure induce Newton-Cartan structures with torsion. In five dimensions the canonical connections contain generalised Coriolis forces, and here also we consider the introduction of torsion by deformation. Non-relativistic limits are exhibited as jumping phenomena of normal bundles. We consider jumping phenomena in Newtonian twistor theory, as well as exhibiting via twistor theory that three-dimensional torsional Newton-Cartan geometry generically exists on the jumping hypersurfaces of Gibbons-Hawking manifolds.