View source for Hermitian structure

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''on a manifold $M$''

A pair $(J,g)$ consisting of a [[Complex structure|complex structure]] $J$ on $M$ and a [[Hermitian metric|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|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|torsion tensor]] $T$ and any vector fields $X$ and $Y$. The [[Curvature tensor|curvature tensor]] $R$ of the canonical connection satisfies the condition $R(JX,JY)=R(X,Y)$. A Hermitian manifold is a [[Kähler manifold|Kähler manifold]] if and only if the canonical Hermitian connection has no torsion and hence is the same as the [[Levi-Civita connection|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|almost-complex structure]] $J$ on $M$ and a [[Riemannian metric|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|Hermitian metric]].