Hermitian metric

A Hermitian metric on a complex vector bundle over a complex manifold is a smoothly varying family of Hermitian inner products on the fibers of the bundle — that is, a positive-definite sesquilinear form that generalizes the notion of a Riemannian metric to the complex setting. Unlike a real metric, which is symmetric, a Hermitian metric satisfies h(v,w) = conjugate(h(w,v)), reflecting the underlying complex structure. Every complex vector bundle admits a Hermitian metric, and the choice of such a metric is the prerequisite for defining the Chern connection, the canonical connection that unifies the bundle's geometric and holomorphic structures.

On the tangent bundle of a complex manifold, a Hermitian metric induces a Riemannian metric on the underlying real manifold, and its imaginary part defines a nondegenerate 2-form. When this 2-form is closed, the metric is called Kähler, and the manifold inherits the full tripartite structure of Kähler geometry. The Hermitian condition is therefore not merely a technical requirement but the bridge between complex linear algebra and differential geometry. Without it, there is no Chern connection, no curvature, and no characteristic classes. The metric is where the geometry begins.