Even if it's easy to find the formal definition of a Riemann surface («a one dimensional complex manifold ...»), I am trying to get an example, as regards the complex logarithm.
In the Wikipedia Riemann surface page, this image is shown:
This seems to be considered the Riemann surface related to the Complex logarithm. In the Complex logarithm page, instead, this same image is captioned as "A plot of the multi-valued imaginary part of the complex logarithm function".
The Complex logarithm imaginary part is a function $f \colon \mathbb{C} \rightarrow \mathbb{R}$ and it is $f = \arg(z)$. This is a multi-level surface, whose layers are vertically spaced by $2 \pi$: layer $n$ has "altitude" $2 \pi n$. The vertical axis (the "altitude") represents the real value(s) assumed by this function for each point $z$ of the complex plane. Assuming that the cut has been performed along the negative real axis, the above image should represent this multi-level surface, which bends (in the 0-level layer) towards the "altitude" $\pi$, when reaching the negative real axis from the $\mathrm{Im(z)} > 0$ half-plane, and bends towards the "altitude" value $-\pi$ when reaching the negative real axis from the $\mathrm{Im(z)} < 0$ half-plane. The edge of layer 0 which has an altitude of $\pi$ perfectly joins the edge of layer 1 which has an altitude $-\pi$ (being at layer 1, it has an actual altitude $2\pi - \pi = \pi$).
Question 1: Are the Riemann surface of the Complex logarithm and the imaginary part of the Complex logarithm the same thing?
My attempt: Not exactly. As far as I understood, the Riemann surface for the Complex logarithm has only the same "shape" as $\arg(z)$. Each layer is cut along the negative real axis, thus generating two edges; one edge joins the corresponding edge of the above layer and the other edge joins the corresponding edge of the below layer. But the layers are not separated by a $2 \pi$ distance: they are all overlapped with no vertical spacing between them.
Question 2: Is this correct?