  We work on a parallelizable time-orientable Lorentzian 4-manifold and prove
that in this case the notion of spin structure can be equivalently defined in a
purely analytic fashion. Our analytic definition relies on the use of the
concept of a non-degenerate two-by-two formally self-adjoint first order linear
differential operator and gauge transformations of such operators. We also give
an analytic definition of spin structure for the 3-dimensional Riemannian case.
