Non-Relativistic Twistor Theory and Newton–Cartan Geometry

We develop a non–relativistic twistor theory, in which Newton–Cartan structures of Newtonian gravity correspond to complex three–manifolds with a four–parameter family of rational curves with normal bundle O⊕O(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O} \oplus \mathcal {O}(2)}$$\end{document}. We show that the Newton–Cartan space-times are unstable under the general Kodaira deformation of the twistor complex structure. The Newton–Cartan connections can nevertheless be reconstructed from Merkulov’s generalisation of the Kodaira map augmented by a choice of a holomorphic line bundle over the twistor space trivial on twistor lines. The Coriolis force may be incorporated by holomorphic vector bundles, which in general are non–trivial on twistor lines. The resulting geometries agree with non–relativistic limits of anti-self-dual gravitational instantons.