  Defining a spin connection is necessary for formulating Dirac's bispinor
equation in a curved space-time. Hestenes has shown that a bispinor field is
equivalent to an orthonormal tetrad of vector fields together with a complex
scalar field. In this paper, we show that using Hestenes' tetrad for the spin
connection in a Riemannian space-time leads to a Yang-Mills formulation of the
Dirac Lagrangian in which the bispinor field is mapped to a set of Yang-Mills
gauge potentials and a complex scalar field. This result was previously proved
for a Minkowski space-time using Fierz identities. As an application we derive
several different non-Riemannian spin connections found in the literature
directly from an arbitrary linear connection acting on Hestenes' tetrad and
scalar fields. We also derive spin connections for which Dirac's bispinor
equation is form invariant. Previous work has not considered form invariance of
the Dirac equation as a criterion for defining a general spin connection.
