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Differential Geometry Question

  1. Nov 20, 2009 #1
    1. The problem statement, all variables and given/known data
    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]?

    2. Relevant 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]


    3. 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?
     
  2. jcsd
  3. Nov 20, 2009 #2

    Dick

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    They have to mean the eigenvalues of the derivative of the Gauss map, dN, the mapping between the tangent spaces of the plane and the torus.
     
  4. Nov 20, 2009 #3
    Perhaps your right. The image of the Gauss map at a point is perpendicular to the tangent space at that point , so that no linear combination of [tex]\partial_1\sigma[/tex], [tex]\partial_2\sigma[/tex] could ever represent N at that point. However is it possible to represent the map N:R2->R3 as a matrix? I'll try to find out if there is an error in the question.
     
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