A Kähler structure on a complex manifold M combines a Riemannian metric on the underlying real manifold with the complex structure. Such a structure brings together geometry and complex analysis, and the main examples come from algebraic geometry. When M has n complex dimensions, then it has 2n real dimensions. A Kähler structure is related to the unitary group U(n), which embeds in S O(2n) as the orthogonal matrices that preserve the almost complex structure (multiplication by 'i'). In a coordinate chart, the complex structure of M defines a multiplication by i and the metric defines orthogonality for tangent vectors. On a Kähler manifold, these two notions (and their derivatives) are related.

complex manifold | complex structure | Kähler form | Kähler identities | Kähler manifold | Kähler metric | Kähler potential | Riemann surface | symplectic manifold