Hermitian metric
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A hermitian metric on a complex vector bundle E over a smooth manifold M, is a positive-definite, hermitian inner product on each fiber Ep, that varies smoothly with the point p in M.
An important special case is that of a hermitian metric on the complexified tangent bundle <math> TM \otimes \mathbb C <math> of a complex manifold M. This is the hermitian analogue of a Riemannian metric. Every complex manifold admits such a hermitian metric.
See also: Kähler metric