Complex Geometry/Holomorphic manifolds

holomorphic manifold A holomorphic manifold is a differentiable manifold whose transition maps are holomorphic. Riemann sphere Consider the set 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 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.