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Complex Geometry/Holomorphic manifolds

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Definition (holomorphic manifold):

A holomorphic manifold is a differentiable manifold whose transition maps are holomorphic.

Example (Riemann sphere):

Consider the set

Theorem (Riemann's extension theorem):

Let be a complex manifold, let be holomorphic, and let be holomorphic, such that for all and all , the function is bounded on . Then there exists a unique function that extends .

Exercises

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  1. Consider the stereographic projection from a sphere to the complex plane, where the sphere is of radius and tangent to the zero point of the complex plane. Prove that the function corresponds to a reflection of the sphere about the equator.