  Recently the vector inflation has been proposed as the alternative to
inflationary models based on scalar bosons and quintessence scalar fields. In
the vector inflationary model, the vector field non-minimally couples to
gravity. We should, however, inquire if there exists a relevant fundamental
theory which supports the inflationary scenario. We investigate the possibility
that Weyl's gauge gravity theory could be such a fundamental theory. That is
the reason why the Weyl's gauge invariant vector and scalar fields are
naturally introduced. After rescaling the Weyl's gauge invariant Lagrangian to
the Einstein frame, we find that in four dimensions the Lagrangian is
equivalent to Einstein-Proca theory and does not have the vector field
non-minimally coupled to gravity, but has the scalar boson with a polynomial
potential which leads to the spontaneously symmetry breakdown.
