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

Mahsa Beizaei
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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 $\{(x_1,x_2,...,x_n) \in \mathbb{R}^n : x_n \geq 0\}$

allizdog
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  • I would note that the term "manifold with boundary" usually allows the boundary to be empty, so the Riemann sphere would still be a complex manifold with boundary. – Eric Wofsey Nov 16 '16 at 07:37
  • What is the definition of a complex manifold with boundary? – Mahsa Beizaei Nov 16 '16 at 07:46
  • http://math.stackexchange.com/questions/1957840/what-is-the-definition-of-a-complex-manifold-with-boundary – Eric Wofsey Nov 16 '16 at 07:50
  • yes, I ment the boundary is empty, but usually we say that a manifold has a boundary if it has a non-empty boundary – allizdog Nov 16 '16 at 07:55