Hermitian structure

on a manifold $M$

A pair $(J,g)$ consisting of a complex structure $J$ on $M$ and a Hermitian metric $g$ in the tangent space $TM$, that is, a Riemannian metric $g$ that is invariant under $J$:

$$g(JX,JY)=g(X,Y)$$

for any vector fields $X$ and $Y$ on $M$. A Hermitian structure specifies in any tangent space $T_pM$ the structure of a Hermitian vector space (see Hermitian metric). A manifold with a Hermitian structure is called a Hermitian manifold. A Hermitian structure defines on $M$ a differential $2$-form $\omega(X,Y)=g(X,JY)$, which is called the canonical $2$-form of the Hermitian manifold. Any structure $J$ on $M$ can be completed by some Riemannian metric $g$ to a Hermitian structure $(J,g)$: for $g$ one can take the metric $g(X,Y)=g_0(X,Y)+g_0(JX,JY)$, where $g_0$ is an arbitrary metric. The canonical Hermitian connection of a Hermitian metric $g$ can be regarded as an affine connection with torsion $T$ on $M$ relative to which the fields $J$ and $g$ are parallel. Among all affine connections satisfying these conditions it is uniquely characterized by the identity $T(JX,Y)=T(X,JY)$, which is valid for its torsion tensor $T$ and any vector fields $X$ and $Y$. The curvature tensor $R$ of the canonical connection satisfies the condition $R(JX,JY)=R(X,Y)$. A Hermitian manifold is a Kähler manifold if and only if the canonical Hermitian connection has no torsion and hence is the same as the Levi-Civita connection of $g$.

A natural generalization of the concept of a Hermitian structure is that of an almost-Hermitian structure, which is a pair $(J,g)$ consisting of an almost-complex structure $J$ on $M$ and a Riemannian metric $g$ that is invariant under $J$. If the fundamental $2$-form $\omega(X,Y)=g(X,JY)$ is closed, then an almost-Hermitian structure is called almost Kählerian. The specification of an almost-Hermitian structure is equivalent to a reduction of the structure group of the tangent bundle to the group $U(n)$, where $n=\dim M/2$. Any non-degenerate differential $2$-form on a manifold $M$ is the fundamental $2$-form of some almost-Hermitian structure.

For references see Hermitian metric.