A Hermitian metric on a complex vector bundle assigns a Hermitian inner product to every fiber bundle. The basic example is the trivial bundle π:U×C^k->U, where U is an open set in R^n. Then a positive definite Hermitian matrix H defines a Hermitian metric by 〈v, w〉 = v^T Hw^_, where w^_ is the complex conjugate of w. By a partition of unity, any complex vector bundle has a Hermitian metric.
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