# p-brane Newton--Cartan Geometry

## TL;DR

This paper formalizes p-brane Newton--Cartan geometry, establishing foundational conditions for torsion-free, compatible affine connections, and generalizes classical Newton--Cartan and Riemannian geometries to p-brane contexts.

## Contribution

It provides a rigorous definition of p-brane Newton--Cartan geometry and characterizes conditions for compatible affine connections, extending Newton--Cartan and Riemannian geometries.

## Key findings

- Necessary and sufficient conditions for torsion-free, compatible affine connections.
- Characterization of the space of such affine connections.
- Generalization of Newton--Cartan and Riemannian geometries.

## Abstract

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 non-relativistic 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. Since pNC structures interpolate between Leibnizian structures for p=0 and Lorentzian structures for p=d-1 (with d the dimension of the spacetime manifold), the present work also constitutes a generalisation of results of Newton--Cartan and (pseudo-) Riemannian geometry.