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Homework Help: Top/mani torus T^2=R^2/Z^2

  1. Mar 1, 2015 #1
    1. The problem statement, all variables and given/known data

    Consider the function f= sin(4pix)cos(6pix) on torus T^2=R^2/Z^2

    a) prove this is a morse function and calculate min, max, saddle.

    b) describe the evolution of sublevel sets f^-1(-inf, c) as c goes from min to max
    2. Relevant equations
    grad(f)= <partial x, partial y>

    show hessian matrix not equal to zero

    3. The attempt at a solution

    From what I understand

    1st need to find critical points. so take grad and set equal to zero

    2nd use hessian matrix with those critical values that i found before and see if non zero

    BUT, i dont know what torus T^2=R^2/Z^2 looks like. What does the T^2 mean? I believe R^2/Z^2 is just the xy graph because z has been removed. so its like 3D but if remove z then 2D

    so is this a square flat torus?

    once I know the shape then I can do the part b part since all you have to do is fill the shape with "water" and see how the topology changes within the critical values.

    So is this correct? Since its cos and sin how do i know which critical values to pick and are within the domain.
  2. jcsd
  3. Mar 2, 2015 #2


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    2017 Award

    Staff: Mentor

    I would interpret that notation as [0,1] x [0,1] where the edges are identified with each other, in the same way other objects are defined with the X/Y notation.

    Everything that is in your torus is relevant. The functions have a period of 1 (and a smaller one but that is not important for this point), so identifying -0.8, 1.2, ... with .2 for example works nicely.
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