I recently asked a question in my mathematical physics class: In complex manifolds, why don't we combine the real part of one variable with the imaginary part of the other variable to obtain more general CR-equations? My lectures mentioned something about conformal manifolds. But I didn't understand quite what he was talking about. Could you please refer me to literature which deals with the question that I put? I haven't read any literature on Functions of Several Complex Variables, but I could imagine that they the CR equations could look something like this: $$\frac{\partial u}{\partial x_i}=\frac{\partial v}{\partial y_j}$$ $$\frac{\partial u}{\partial y_i}=-\frac{\partial v}{\partial x_j}$$ where i and j are natural numbers and run from 1 to dimension of the complex space.
thank you
Choose your favorite complex manifold. In differential geometry, there is often a source manifold (call it a domain if you like) and a target manifold (call it the image of the pre-image domain, or if you like, the range of a map from a source to target). Conformal simply means angles in the pre-image of the map are preserved in the target under the conformal map.
I found Theodore W. Gamelin, "Complex Analysis," Springer, 2001, page 59 helpful as it illustrates with a picture a conformal map.
For the most part, I believe John Ma is giving you good advice.