Hermitian manifold - Imported Knowlpedia Content

Definition

Let $M$ be a complex manifold with its canonical complex structure $J$ on the real tangent bundle. A Hermitian manifold is $M$ together with a smooth Riemannian metric $g$ satisfying

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

for all real tangent vectors $X,Y$ at the same point. Thus $(M,J,g)$ is an almost-Hermitian manifold whose almost-complex structure is integrable. Equivalently, $g$ determines a Hermitian metric on the complex tangent bundle. Its fundamental form is the real $2$ -form $\omega(X,Y)=g(JX,Y)$ .

Associated form and local data

The fundamental $2$ -form is nondegenerate and of complex type $(1,1)$ . In holomorphic coordinates, the metric is represented by a smoothly varying positive-definite Hermitian matrix. These descriptions carry the same data as the $J$ -invariant real Riemannian metric; they do not identify that real bilinear form with a sesquilinear form.

Every complex manifold under the usual second-countable convention admits a Hermitian metric: local standard metrics can be combined with a smooth partition of unity.

Relationship to Kähler geometry

Hermitian compatibility does not require $d\omega=0$ . A Hermitian manifold is Kähler precisely when its fundamental form is closed. This separates integrability of $J$ , already built into the complex-manifold hypothesis, from the additional differential condition on $\omega$ ; see Huybrechts, §3.1.

Examples and non-examples

Complex Euclidean space with its standard Euclidean metric is Hermitian, as is complex projective space with the Fubini–Study metric. On $\mathbb C^n$ , a generic Riemannian metric is not Hermitian: if it assigns different lengths to $X$ and $JX$ , it fails $J$ -invariance.

References

Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. DOI record. Relevant: §3.1, especially Definition 3.1.1 for the fundamental form of a Hermitian metric.