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Hermitian manifold

From Wikipedia, the free encyclopedia

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

g_{pq} = J^r_p J^s_q g_{rs}

A manifold with such a metric is said to be a Hermitian manifold or to possess a Hermitian structure.

[edit] Hermitian form

The differential form

\omega(\cdot, \cdot) = g(J\cdot, \cdot)

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

\omega_{rq} = J_r^p g_{pq}

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

\nabla J=0

and

\nabla \omega = 0

where \nabla 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.

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