Leibnizian, Galilean and Newtonian structures of spacetime

The following three geometrical structures on a manifold are studied in detail: (1) Leibnizian: a non-vanishing 1-form $\mathrm{\Omega }$ plus a Riemannian metric $\text{h}$ on its annhilator vector bundle. In particular, the possible dimensions of the automorphism group of a Leibnizian G-structure are characterized. (2) Galilean: Leibnizian structure endowed with an affine connection $\mathrm{\nabla }$ (gauge field) which parallelizes $\mathrm{\Omega }$ and $\text{h}$. Fixed any vector field of observers Z ($\mathrm{\Omega }(Z)=1$), an explicit Koszul–type formula which reconstruct bijectively all the possible $\mathrm{\nabla }$’s from the gravitational $\mathcal{G}={\mathrm{\nabla }}_{Z}Z$ and vorticity $\omega =rotZ\mathrm{/}2$ fields (plus eventually the torsion) is provided. (3) Newtonian: Galilean structure with $\text{h}$ flat and a field of observers Z which is inertial (its flow preserves the Leibnizian structure and $\omega =0$). Classical concepts in Newtonian theory are revisited and discussed.