In this chapter, we will use the principal bundle description of Galilei manifolds to introduce the notion of so-called Bargmann forms. These allow for a global formulation of the classification theorem for Galilei connections, i.e. one that is independent of a choice of unit timelike vector field. We will also see how the formalism of Bargmann forms enables a description of the coupling of massive matter to Newton–Cartan gravity via variational principles, and how it allows for a more general and computationally easier description of the limit from GR to Newton–Cartan gravity than that we encountered before. Finally, we will discuss how the quotient of a Lorentzian manifold by a lightlike symmetry naturally carries the structure of a Galilei manifold.