Is the Riemann sphere complex manifold with boundary?

Question

Can anybody help me? I know that, Is the riemannian sphere complex manifold with boundary and is compact complex manifold?

Answer 1

The Riemann sphere is indeed a compact complex manifold but it does not have boundary if that's what you are asking. It's the compactification of the complex plane. You can look at it as taking a closed ball in the complex plane and contracting its boudary to a single point obtaining a compact manifold with no boundary. In the book "introduction to general topology " by Sze-tsen Hu, the compatification is well explained. The stereographic projections give you parametrisations of the riemann sphere.

The definition of a manifold with boundary is an hausdorff topological space with countable basis which is locally homeomorphic to $\{(x_1,x_2,...,x_n) \in \mathbb{R}^n : x_n \geq 0\}$