# Kähler metric

A Hermitian metric on a complex manifold whose fundamental form $\omega$ is closed, i.e. satisfies the condition $d \omega = 0$. Examples: the Hermitian metric $\sum _ {k = 1 } ^ {n} | dz _ {k} | ^ {2}$ in $\mathbf C ^ {n}$; the Fubini–Study metric on the complex projective space $\mathbf C P ^ {n}$; and the Bergman metric (see Bergman kernel function) in a bounded domain in $\mathbf C ^ {n}$. A Kähler metric on a complex manifold induces a Kähler metric on any submanifold. Any Hermitian metric on a one-dimensional manifold is a Kähler metric.
A Hermitian metric $h$ on a complex manifold is a Kähler metric if and only if it satisfies any one of the following conditions: parallel transfer along any curve (relative to the Levi-Civita connection) is a complex linear mapping, i.e. it commutes with the complex structure operator; the complex Laplacian $\square$ corresponding to the metric $h$ on differential forms satisfies the condition $\overline \square \; = \square$, i.e. the Laplace operator $\Delta$ is precisely $2 \square$; local coordinates can be introduced in a neighbourhood of any point, relative to which the matrix of $h$ coincides with the identity matrix up to second-order quantities (see , ).