Leibnizian, Galilean and Newtonian structures of space-time

The following three geometrical structures on a manifold are studied in detail: Leibnizian: a nonvanishing one-form Omega plus a Riemannian metric <.,.> on its annhilator vector bundle. In particular, the possible dimensions of the automorphism group of a Leibnizian G-structure are characterized. Galilean: Leibnizian structure endowed with an affine connection del (gauge field) which parallelizes Omega and <.,.>. For any fixed vector field of observers Z(Omega(Z)equivalent to1), an explicit Koszul-type formula which reconstructs bijectively all the possible del's from the gravitational G:=del(Z)Z and vorticity omega:= 1/2 rot Z fields (plus eventually the torsion) is provided. Newtonian: Galilean structure with <.,.> flat and a field of observers Z which is inertial (its flow preserves the Leibnizian structure and omegaequivalent to0). Classical concepts in Newtonian theory are revisited and discussed. (C) 2003 American Institute of Physics.