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# Kähler Manifold

A complex manifold for which the exterior derivative of the fundamental form associated with the given Hermitian metric vanishes, so . In other words, it is a complex manifold with a Kähler structure. It has a Kähler form, so it is also a symplectic manifold. It has a Kähler metric, so it is also a Riemannian manifold.

The simplest example of a Kähler manifold is a Riemann surface, which is a complex manifold of dimension 1. In this case, the imaginary part of any Hermitian metric must be a closed form since all 2-forms are closed on a two real dimensional manifold.

Calibrated Manifold, Complex Manifold, Complex Projective Space, Hyper-Kähler Manifold, Kähler Form, Kähler Identities, Kähler Metric, Kähler Potential, Kähler Structure, Projective Algebraic Variety, Quaternion Kähler Manifold Riemannian Metric, Symplectic Manifold

Portions of this entry contributed by Todd Rowland

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## Cite this as:

Rowland, Todd and Weisstein, Eric W. "Kähler Manifold." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KaehlerManifold.html