I have the following question. Provided that $E\to M$ is vector bundle and that a Clifford module $Cl(T^*M)$ acts on $\Gamma(E)$ via Clifford multiplication $c$, the Dirac operator of this Clifford bundle is defined to be $$D = c \circ \nabla$$

If $x\in M$ and $e_1,\dots,e_n$ a local frame in $x\in U\subseteq M$, then the local form of the Dirac operator is said to be $$D = \sum_i c(e^i) \nabla_i$$

The question is: This equality holds everywhere in the local area $U$ or only at the origin? And if it holds only at $x$ why do we say "local can be written".

If we take a synchronous frame around $x$ (we extend $e_1,\dots, e_n \in T_pM$ by parallel transport, along radial lines in a small area around $x$) then we can have that

$$\nabla s = \sum d(f_i) e_i + \sum f_i \nabla e_i$$

for $s = \sum f_i e_i$, but the second terms is zero only at $x$. It doesn't seem to hold in all points around $x$.

Anyone who can help?