Hermitian manifold
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In mathematics, a Hermitian manifold is a smooth manifold with a metric g which is Hermitian for an almost complex structure J.
[edit] Definition
A metric g is called a Hermitian metric if
- g(v,w) = g(Jv,Jw)
for all vector fields v, w on M. In index notation, one writes
A manifold with such a metric is said to be a Hermitian manifold or to possess a Hermitian structure.
[edit] Hermitian form
is called the Hermitian form of J. That is, for all vector fields u, v on M, one has
- ω(u,v) = g(Ju,v)
In index notation, one writes
The form ω is a form of type-(1,1), indicating that it has a holomorphic and an anti-holomorphic part. The decomposition of differential forms into holomorphic and anti-holomorphic subspaces is discussed in the article on almost-complex manifolds.
[edit] Kähler form
If the Hermitian form is closed, dω = 0, then the form is called a Kähler form, and the Hermitian manifold is called a Kähler manifold. Conditions equivalent to the vanishing of dω = 0 are
and
where 
is the Levi-Civita connection of g. The last two conditions imply that J and ω are constant tensors on the manifold, and thus are preserved by the holonomy group of M. In essence, this implies that the holonomy group of a Kähler manifold is a subgroup of U(m), where the dimension of M is 2m.
A Kähler form is a symplectic form, and thus, Kähler manifolds are naturally symplectic manifolds.
