electron wave equation

I have noticed that if a curved space-time M is equipped with a
parallelization w: T(M) -> R^4, and the Dirac lagrangean is formulated
as,

L = Psi^bar (i gamma^k (d Psi) /\ * w_k + (i/2) gamma^k Psi d * w_k -
- m Psi * 1)

(Psi^bar = Psi^h gamma^0, w_k = eta_kl w^l and w^l is a component of
w), then extremizing the action by varying Psi in the usual manner
leads to the wave equation,

i gamma^k (d Psi) /\ * w_k + (i/2) gamma^k Psi d * w_k - m Psi * 1 = 0

and the current,

j = Psi^bar gamma^k Psi w_k

has the property d * j = 0.

.