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what is the actual definition of "complex coordinate system"?

I am not referring for 'complex number' nor 'polar coordinate system'.

These terms are overlapping in my mind and i am unable to get what "complex coordinate system" is?

Please help me , this is the very first time i used any forum for solving my problem.

Looking forward for your helps.

Thanks in advance

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you have too precise for which vector space over which field ? –  Abdelmajid Khadari Mar 31 '12 at 15:14
    
In what context did you encounter this term? –  Hans Lundmark Mar 31 '12 at 15:16

1 Answer 1

When you do real differential geometry on a $d$-dimensional manifold $M$ you have an atlas of local coordinates $(x_1,\ldots, x_d)$, where each individual chart of this atlas provides a parametrization of a small part ("coordinate patch") of $M$ as a copy of, say, the unit ball in ${\mathbb R}^d$.

Now when you do complex geometry on a complex manifold $M$ which is not simply a domain of ${\mathbb C}$ then you also need an atlas. When the complex dimension of $M$ is just $1$ then each individual chart of this atlas provides a parametrization of a small part of $M$ as a copy of, say, the unit disk in ${\mathbb C}$. This means that in the neighborhood of each point $p\in M$ at least one complex coordinate function $z(\cdot)$ is defined which maps this neighborhood bijectively on the unit disk of the $z$-plane, and if a point $p$ happens to lie in two different such charts $z(\cdot)$ and $w(\cdot)$ then the two "variables" $z$ and $w$ are related in a biholomorphic way: $w=\phi(z)$, where $\phi$ is holomorphic in a neighborhood of $0$, and $\phi'(z)\ne0$ where it is defined.

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very nice. Can you please also see this thread, math.stackexchange.com/questions/127555/… I will be very thankful to you. –  user960567 Apr 3 '12 at 13:25

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