A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.
What I interpret from this is that any function that takes on complex values on its domain is always differentiable (in the complex field).
Given two manifolds $M$ and $N$, a differentiable map $f : M → N$ is called a diffeomorphism if it is a bijection and its inverse $f^{−1} : N → M$ is differentiable as well.
Here it is essentially the same as a holomorphism except it takes on real values instead of complex values (because manifolds resemble Euclidean space at any point on it).
Thus holomorphic functions and diffeomorphic functions are the same, except holomorphic functions have domain $\Bbb{C}$ and diffeomorphic functions have domain $\Bbb{R}$.
Am I correct?
Best Answer
The two notions of being differentiable used here, $\mathbb R$-differentiability and $\mathbb C$-differentiability, are not the same. The definitions may look similar, but the implications of these definitions are very different.
A diffeomorphism is required to have a nonvanishing derivative. The derivative of a holomorphic function is allowed to be zero.
A diffeomorphism is required to be a bijection. A holomorphic function does not have to be.