  Motivated by an axiomatic approach to characterize space-time it is
investigated a reformulation of Einstein's gravity where the pseudo-riemannian
geometry is substituted by a Weyl one. It is presented the main properties of
the Weyl geometry and it is shown that it gives extra contributions to the
trajectories of test particles, serving as one more motivation to study general
relativity in Weyl geometry. It is introduced its variational formalism and it
is established the coupling with other physical fields in such a way that the
theory acquires a gauge symmetry for the geometrical fields. It is shown that
this symmetry is still present for the red-shift and it is concluded that for
cosmological models it opens the possibility that observations can be fully
described by the new geometrical scalar field. It is concluded then that this
reformulation, although representing a theoretical advance, still needs a
complete description of their objects.
