- #1
Sistine
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Homework Statement
Consider the following parametrization of a Torus:
[tex]\sigma(u,v)=((R+r\cos u)\cos v, (R+r\cos u)\sin v, r\sin u)[/tex]
[tex] R>r,\quad (u,v)\in [0,2\pi)^2[/tex]
1. Compute the Gauss map at a given point.
2. What are the eigenvalues of that map in the base [tex](\partial_1\sigma,\partial_2\sigma)[/tex]?
Homework Equations
[tex]\partial_1\sigma=\frac{\partial\sigma}{\partial u}[/tex]
[tex]\partial_2\sigma=\frac{\partial\sigma}{\partial v}[/tex]
The Gauss map is defined as:
[tex]N(u,v)=\frac{\partial_1\sigma\times\partial_2\sigma}{|\partial_1\sigma\times\partial_2\sigma|}[/tex]
The Attempt at a Solution
Computing the Gauss map at a point [tex]p[/tex] is straightforward enough. But I'm not sure what part 2 of the question is asking me to do. How can I visualize the map as a matrix operator in a certain basis so that I can compute its eigenvalues?