Top/mani torus T^2=R^2/Z^2

[Fellowroot]

Homework Statement

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

Homework Equations

grad(f)= <partial x, partial y>

show hessian matrix not equal to zero

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 don't 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.

 

[mfb]

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.

Since its cos and sin how do i know which critical values to pick and are within the domain.

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.