f(R) gravity, torsion and non-metricity
  For both f(R) theories of gravity with an independent symmetric connection
(no torsion), usually referred to as Palatini f(R) gravity theories, and for
f(R) theories of gravity with torsion but no non-metricity, called U4 theories,
it has been shown that the independent connection can actually be eliminated
algebraically, as long as this connection does not couple to matter.
Remarkably, the outcome in both case is the same theory, which is dynamically
equivalent with an \omega_0=-3/2 Brans--Dicke theory. It is shown here that
even for the most general case of an independent connection with both
non-metricity and torsion one arrives at exactly the same theory as in the more
restricted cases. This generalizes the previous results and explains why
assuming that either the torsion or the the non-metricity vanishes ultimately
leads to the same theory. It also demonstrates that f(R) actions cannot support
an independent connection which carries dynamical degrees of freedom,
irrespectively of how general this connection is, at least as long as there is
no connection-matter coupling.
