I am trying to describe the Riemann surface $F_{\lambda}$ corresponding to the function $w=\sqrt{\lambda-z^2}$ where $\lambda\in \mathbb{C}, \lambda\ne 0$ over $|z|\le 2$. What I have so far are the following:
Its branch points are $\pm \sqrt{\lambda}$. By removing a branch cut that connects $-\sqrt{\lambda}$ and $\sqrt{\lambda}$, we obtain a domain $D$ of $z$ so that two copies of $D$ correspond to two single-valued functions on $D$ coming from the function $w=\sqrt{\lambda-z^2}$, where one is negative of the other at the same input $z$. These two copies of $D$ are open subsets of $F_{\lambda}$.
To glue these two copies (drawn as two squares on the left of figure), observe that if we analytically continued $w$ on one copy of $D$ along a closed path $\Delta$ that encloses the branch cut, we still stay in the same $D$. However, if analytically continued $w$ along $\nabla$ that passes through the branch cut (e.g. we let $z$ runs from $-2i$ to $2i$ in the figure), we will move from one $D$ to another as soon as we pass the branch cut. Therefore, if we open up the branch cut into $\Delta$, the two copies can be glued along $\Delta$ to create a cylinder as in figure. Note that in order to match $\nabla$ correctly and to match the branch points $\pm \sqrt{\lambda}$ when glueing two copies, we first need to flip one copy of $D$ across the real axis, as indicated in the figure.
My question is: Say I want to assign an orientation to $F_{\lambda}$ as in the figure, did I get the correct induced orientation on $D$ as in the figure? Or in other words, am I being consistent with my choice of orientation in the figure?
Any help would be much appreciated!
Briefly, the orientation in your diagrams looks correct. In case it's of interest, it's possible to depict the Riemann surface in real three-space by graphing the real parts of the branches of square root. The blue mesh is the sheet where the real part is positive in the upper half-plane and negative in the lower half-plane; the slit from $-\sqrt{\lambda}$ to $\sqrt{\lambda}$ is opened into a literal circle. The gray-shaded surface is the other sheet. The $z$-plane is shown by its real and imaginary axes; vertical projection is the two-sheeted branched cover.