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Math Help - Gauss bonnet theorem

  1. #1
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    Gauss bonnet theorem

    For this surface M, I guess I need to show M has no boundary, and find the euler characterstic of M. How to do this?
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  2. #2
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    Re: Gauss bonnet theorem

    You can show that M is homeomorphic to S^2 so it has no boundary and its Euler character is 2.
    To show that M is homeomorphic to S^2={(x,y,z)|x^2+y^2+z^2=1}, define a map: f: M -> S^2 by f(p) = p/|p| and try to show that f is a homeomorphism( actually it is a diffeomorphism).
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  3. #3
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    Re: Gauss bonnet theorem

    Ok thanks, so the total curvature is 2 x 2pi=4pi. What about (ii)
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  4. #4
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    Re: Gauss bonnet theorem

    (ii) is a direct application of the theorem. -4 pi = 2 pi X, so X=-2. It is a double torus.
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