  The following three geometrical structures on a manifold are studied in
detail: (1) Leibnizian: a non-vanishing 1-form $\Omega$ plus a Riemannian
metric $\h$ 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 $\nabla$ (gauge field) which parallelizes $\Omega$ and $\h$. Fixed
any vector field of observers Z ($\Omega (Z) = 1$), an explicit Koszul--type
formula which reconstruct bijectively all the possible $\nabla$'s from the
gravitational ${\cal G} = \nabla_Z Z$ and vorticity $\omega = rot Z/2$ fields
(plus eventually the torsion) is provided. (3) Newtonian: Galilean structure
with $\h$ 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.
