Critical and regular values of a map from $T^2$ to $\mathbb{R}^2$ Let $T^2$ be the torus in $\mathbb{R^3}$, $\pi:\mathbb{R^3}\rightarrow \mathbb{R^2}$ be the projection $\pi(x,y,z)=(y,z)$ and $p:T^2\rightarrow \mathbb{R^2}$ the restriction of $p$ to $T^2$. I need help finding the critical and regular values of the map $p$. Any assistance is appreciated!
 Definitions: If a differential of a map fails to be surjective at a point then it is a critical point. If a point is in the image of a critical point then it is a critical value. If not, it is a regular value. 
 A: It may be instructive to first solve the simpler problem: What are the critical values of the projection map onto $\mathbb{R}$? This is a common introductory problem in Morse theory, as this is easily visualized as the "height" map. Usually the torus is hung vertically to make the map interesting.
The critical values here are precisely those points which correspond to saddles, basins, or peaks in the torus' geometry. You could, of course, prove this by writing down an actual parameterization, and then the problem would become a question of computing the Jacobian and checking the necessary conditions. But ideally you could use certain theorems to help you out. For example, you might have access to the preimage theorem, which would tell you something about the preimages of your regular values. From there, you could use your geometric intuition about which parts of the projection have such preimages. This would give you a picture of all the regular values, and their complement would be your critical values. Since you've tagged it as "differential topology", and not "advanced calculus", hopefully you have elegant tools such as this, and you don't need to get your hands so dirty.
