  In this paper, we analyse the well-posedness of the initial value formulation
for particular kinds of geometric scalar-tensor theories of gravity, which are
based on a Weyl integrable space-time. We will show that, within a
frame-invariant interpretation for the theory, the Cauchy problem in vacuum is
well-posed. We will analyse the global in space problem, and, furthermore, we
will show that geometric uniqueness holds for the solutions. We make contact
with Brans-Dicke theory, and by analysing the similarities with such models, we
highlight how some of our results can be translated to this well-known context,
where not all of these problems have been previously addressed.
