- #1
mnb96
- 715
- 5
Hello,
I understand the concepts of real differentiable manifold, tangent space, atlas, charts and all that stuff. Now I would like to know how those concepts generalize in the case of a complex manifold.
First of all, what does a coordinate chart for a complex manifold look like? Is it a function [itex]\xi : U\subseteq \mathbb{C}^m \rightarrow \mathbb{C}^n[/itex] where U is a open subset of ℂm ?
Secondly, how do we obtain a tangent space? What are the tangent vectors? In case of real manifolds we obtain them through directional derivatives, but for a complex manifold how do we define a directional derivative?
Thanks.
I understand the concepts of real differentiable manifold, tangent space, atlas, charts and all that stuff. Now I would like to know how those concepts generalize in the case of a complex manifold.
First of all, what does a coordinate chart for a complex manifold look like? Is it a function [itex]\xi : U\subseteq \mathbb{C}^m \rightarrow \mathbb{C}^n[/itex] where U is a open subset of ℂm ?
Secondly, how do we obtain a tangent space? What are the tangent vectors? In case of real manifolds we obtain them through directional derivatives, but for a complex manifold how do we define a directional derivative?
Thanks.