p-brane Newton-Cartan geometry

We provide a formal definition of p-brane Newton-Cartan (pNC) geometry and establish some foundational results. Our approach is the same followed in the literature for foundations of Newton-Cartan gravity. Our results provide control of aspects of pNC geometry that are otherwise unclear when using the usual gauge language of nonrelativistic theories of gravity. In particular, we obtain a set of necessary and sufficient conditions that a pNC structure must satisfy in order to admit torsion-free, compatible affine connections and determine the space formed by the latter. This is summarized in Theorem 3.1. Since pNC structures interpolate between Leibnizian structures for p = 0 and Lorentzian structures for p = d - 1 (with d being the dimension of the spacetime manifold), the present work also constitutes a generalization of results of Newton-Cartan and (pseudo-)Riemannian geometry. Published under license by AIP Publishing.