Let $(E,h) \to M$ be a complex Hermitian vector bundle of rank $r$, with $h$ linear in the first argument and conjugate-linear in the second. Let $U \subset M$ be an [open set](/page/Open%20Set), and let $e=(e_1,\dots,e_r)$ be a local unitary frame for $E|_U$, so that $h(e_i,e_j)=\delta_{ij}$ for all $1 \le i,j \le r$.
paragraph
admin
Let $\nabla$ be a connection on $E|_U$, and define its connection one-form in the frame $e$ to be the matrix $A=(A_{ij})_{1 \le i,j \le r}$ of complex-valued one-forms on $U$ determined by