  In a Minkowski spacetime, one may transform the Dirac wave function under the
spin group, as one transforms coordinates under the Poincar\'e group. This is
not an option in a curved spacetime. Therefore, in the equation proposed
independently by Fock and Weyl, the four complex components of the Dirac wave
function transform as scalars under a general coordinate transformation. Recent
work has shown that a covariant complex four-vector representation is also
possible. Using notions of vector bundle theory, we describe these two
representations in a unified framework. We prove theorems that relate together
the different representations and the different choices of connections within
each representation. As a result, either of the two representations can account
for a variety of inequivalent, linear, covariant Dirac equations in a curved
spacetime that reduce to the original Dirac equation in a Minkowski spacetime.
In particular, we show that the standard Dirac equation in a curved spacetime,
with any choice of the tetrad field, is equivalent to a particular realization
of the covariant Dirac equation for a complex four-vector wave function.
