Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Here is another attempt, which I think should work. If it is, I'll open another question asking for reference. Let $f$ be a continuous map from 3-ball $\mathbb{B}^3$ to $\mathbb{R}^2$, and $C$ be a closed curve in $\partial \mathbb{B}^3$. If $f(C)$ has a non-zero winding number around $0$, then the degree theory tell me that $f^{-1}(0)$ has a non-empty intersection with any disk in $\mathbb{B}^3$ bounded by $C$. If $f$ is moreover real analytic, then $f^{-1}(0)$ contains a path-connected curve that intersects every disk bounded by $C$ (is there a word for this situation?)
@OlegEroshkin You are right. As I regard $\mathbb{S}^2$ as compactification of $\mathbb{R}^2$, I mean the winding number of $f(C)$ in $\mathbb{S}^2 \setminus \infty$. It is now clarified in the question.
FYI, I consulted with the wikipedia editor. He used a software to calculate the subgroups of the tetrahedral group. He also mentioned Conway's book "the symmetries of things". There is a poset for the octahedral group, from which one can extract a poset for the tetrahedral group (if he manages to understand the notation system).
@MohammadGhomi The things you mention here are all much simpler without curvature, by noticing that the map from lifting to curvature is continuous. For example: when you say "rescaling" you mean rescaling the height; when you say open, you mean the interior of a secondary cone. These topological statements are all consequences of the secondary cone / fan, and are much nicer with lifting instead of Gaussian curvature. If you insist, I suggest looking at mean curvature (dihedral angle $\times$ edge length), which might behave nicer. Although still no more info than the secondary cone.
