Almost-Hermitian manifold - Imported Knowlpedia Content

Definition

An almost-Hermitian manifold is a triple $(M,J,g)$ in which $M$ is a smooth manifold, $J$ is an almost-complex structure, and $g$ is a Riemannian metric satisfying

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

for all tangent vectors $X,Y$ at the same point. Equivalently, $g$ is the real part of a Hermitian metric on the complex vector bundle $(TM,J)$ . The associated fundamental $2$ -form is

\omega(X,Y)=g(JX,Y).

It is smooth and nondegenerate, but it need not be closed.

Compatible triples

Any two of $J$ , $g$ , and $\omega$ determine the third when they satisfy the compatibility and positivity conditions. In particular,

g(X,Y)=\omega(X,JY)

for the sign convention in the core. The linear-algebra construction can be carried out smoothly, and every almost-complex manifold admits compatible metrics; see Cannas da Silva, “Compatible Almost Complex Structures”.

Relationship to symplectic and complex geometry

If $d\omega=0$ , then $(M,\omega)$ is a symplectic manifold and the structure is called almost Kähler. If $J$ is integrable, the manifold is Hermitian. Requiring both integrability and $d\omega=0$ gives a Kähler structure. Neither condition follows from almost-Hermitian compatibility alone.

Examples and conventions

Euclidean space $\mathbb R^{2n}$ , with its standard complex structure and Euclidean metric, is almost Hermitian. More generally, a Hermitian metric on a complex manifold gives an almost-Hermitian structure on its underlying smooth manifold. Some authors define the fundamental form as $g(X,JY)$ , which is the negative of the convention used here.

References

Ana Cannas da Silva, Lectures on Symplectic Geometry, Springer, 2008. DOI record. Relevant: “Almost Complex Structures” and “Compatible Triples.”
Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. DOI record. Relevant: chapter on Kähler manifolds and Hermitian structures.