Hermitian manifold

Remarks

Example: $\C^n = \R^{2n}$ with the Euclidean metric $g_0$ and $J$ the multiplication by $i$. Multiplication by $i$ is an orthogonal map of $\R^{2n}$, so $g_0(JX, JY) = g_0(X, Y)$. Its fundamental form is $\omega = \sum_k dx_k \wedge dy_k$, which has constant coefficients, so $d\omega = 0$ and $\C^n$ is Kähler.

Non-example: on $\C = \R^2$ the metric $g = dx^2 + 4\, dy^2$ is Riemannian but not Hermitian. Here $J\partial_x = \partial_y$, so $g(J\partial_x, J\partial_x) = 4 \ne 1 = g(\partial_x, \partial_x)$.

The form $h = g - i\omega$ is a Hermitian form on each tangent space viewed as a complex vector space. Every complex manifold has Hermitian metrics: given any Riemannian metric $g$, the metric $g(X, Y) + g(JX, JY)$ is Hermitian. The same definition with an almost complex structure that need not be integrable gives almost Hermitian manifolds; by the Newlander–Nirenberg theorem the integrable case is the complex case. Every complex submanifold of a Kähler manifold is Kähler with the induced metric.