On the torus, obtained from the unit square $Q=\left\lbrace f(x,y)\in > \mathbb{R}^2: 0\leq x, y \leq 1 \right\rbrace$ by identifying its opposite sides,consider the squared distance of each point from $(1/2; > 1/2): f(x, y) = (x - 1/2)^2 + (y - 1/2)^2$.
(1) Draw level sets of f, highlighting minima, saddles, and maxima. (2) Describe the handle attachments that occurr during the evolution of the sub-level sets of f.
I do understand the questions (point 1 and 2), but I can't understand the introduction. That f is a paraboloid, right? I suppose question 1 and 2 are too easy if f is just that, but how is it the torus involved? I was told to try and solve it graphically but I don't know what I am supposed to draw.
This is what I think the sublevel sets look like, thinking of the torus as a unit square with identified opposite sides:
I believe there are four critical points (one min, one max, two saddle points).
These are different from the ones in the pdf I posted, because they are with respect to a different function.