Smooth unitary frame for a Hermitian connection
= Smooth unitary frame for a Hermitian connection
{title2=$A(V)^\dagger=-A(V)$}
A smooth positive Hermitian bundle metric admits local orthonormal frames by the smooth <Gram-Schmidt process>. In such a frame, metric compatibility gives $A_{ji}(V)+\overline{A_{ij}(V)}=0$ for real <vector fields> $V$, so the connection matrix is skew-Hermitian. The orthonormal frame is smooth; a holomorphic orthonormal frame is not generally available.