= Hermitian metric on a smooth complex vector bundle
{title2=$h_b:E_b\times E_b\to\mathbb C$}
This is a smoothly varying positive-definite <Hermitian form> on each fiber, without requiring a holomorphic base or bundle. A <partition of unity> glues local positive metrics into a global one. Applying smooth Gram-Schmidt to local frames then makes the transitions unitary, reducing the <structure group of a vector bundle> to $U(k)$. For a line bundle its unit vectors form a <principal bundle> for the <circle group>.
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