Hermitian Manifold - Riemannian Metric and Associated Form

Riemannian Metric and Associated Form

A Hermitian metric h on an (almost) complex manifold M defines a Riemannian metric g on the underlying smooth manifold. The metric g is defined to be the real part of h:

The form g is a symmetric bilinear form on TMC, the complexified tangent bundle. Since g is equal to its conjugate it is the complexification of a real form on TM. The symmetry and positive-definiteness of g on TM follow from the corresponding properties of h. In local holomorphic coordinates the metric g can be written

One can also associate to h a complex differential form ω of degree (1,1). The form ω is defined as minus the imaginary part of h:

Again since ω is equal to its conjugate it is the complexification of a real form on TM. The form ω is called variously the associated (1,1) form, the fundamental form, or the Hermitian form. In local holomorphic coordinates ω can be written

It is clear from the coordinate representations that any one of the three forms h, g, and ω uniquely determine the other two. The Riemannian metric g and associated (1,1) form ω are related by the almost complex structure J as follows

for all complex tangent vectors u and v. The Hermitian metric h can be recovered from g and ω via the identity

All three forms h, g, and ω preserve the almost complex structure J. That is,

for all complex tangent vectors u and v.

A Hermitian structure on an (almost) complex manifold M can therefore be specified by either

1. a Hermitian metric h as above,
2. a Riemannian metric g that preserves the almost complex structure J, or
3. a nondegenerate 2-form ω which preserves J and is positive-definite in the sense that ω(u, Ju) > 0 for all nonzero real tangent vectors u.

Note that many authors call g itself the Hermitian metric.