Let $X$ be a complex manifold of complex dimension $n$, and let $(E,h)$ be a Hermitian holomorphic vector bundle of rank $r$ on $X$. Let $h_{\det E}$ denote the Hermitian metric induced by $h$ on the determinant line bundle $\det E=\bigwedge^r E$. With the same Chern-curvature convention used for both $E$ and $\det E$, one has
Equivalently, at any point $x\in X$, choose local holomorphic coordinates $z_1,\dots,z_n$ near $x$ and an $h_x$-unitary basis $e_1,\dots,e_r$ of $E_x$, with [dual basis](/theorems/414) $e^1,\dots,e^r$ of $E_x^*$. If