Complex manifold

In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. More precisely, a complex manifold has an atlas of charts to Cn, such that the change of coordinates between charts are holomorphic.

Related Topics:
Differential geometry - Manifold - Neighborhood - Atlas - Charts - Holomorphic

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Complex manifolds can be regarded as a special case of differentiable manifolds. For example, a 1-dimensional complex manifold is geometrically a surface, known as a Riemann surface. The requirement that the transition functions be holomorphic means that unlike in the general differential case, there is no distinction between different Ck-structures for different k, since holomorphic functions are analytic, and thus any holomorphic structure is also a Ck structure, for any k ≥1.

Related Topics:
Differentiable manifolds - Surface - Riemann surface - Holomorphic - ''C''^''k'' - Structures - Holomorphic functions are analytic

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The theory of complex manifolds has a much different flavor than that of real manifolds, since complex analytic functions are much more rigid than smooth functions. For example, by the Whitney embedding theorem, every real manifold can be embedded as a submanifold of Rn, while it is rare for a complex manifold to be a (complex) submanifold of Cn. Consider for example any compact complex manifold M: any entire function on it must be locally constant, by the extension to several complex variables of Liouville's theorem. This means that M cannot be embedded in Cn unless it has dimension 0. Complex manifolds which can be embedded in Cn (which are necessarily noncompact) are known as Stein manifolds.

Related Topics:
Whitney embedding theorem - Compact - Entire function - Locally constant - Several complex variables - Liouville's theorem - Stein manifold

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One can define an analogue of a Riemannian metric for complex manifolds, called a Kähler metric. Again, unlike the case of real manifolds (which always have Riemannian metrics), it is unusual for a complex manifold to have a Kähler metric.

Related Topics:
Riemannian metric - Kähler metric

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