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Math Help - Gauss-Bonnet help

  1. #1
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    Gauss-Bonnet help

    We found out how to figure out Gauss-Bonnet for spherical geometry in our class, which was using the sum from 1 to n of (alpha + beta + gamma) - pi and turning this into 2piX(x).

    We are supposed to try to find Gauss-Bonnet for hyperbolic and Euclidean. I think I can get it for hyperbolic, it seems very similar to spherical. However, Euclidean he said to work backwards? I'm confused with this, it seems like it is trickier and I don't know where to start.
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  2. #2
    Super Member Rebesques's Avatar
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    Applying the Gauss-Bonnet theorem for Euclidean space is much more simple than its nonzerical counterparts. Consider a triangle with angles a, b, c.
    We know the Gaussian curvature is zero, so all we have to do is sum the jumps of the tangent at each vortex, which ofcourse is \pi-a+\pi-b+\pi - c=3\pi-a-b-c. So we end up with 2\pi-(3\pi-a-b-c)=0 or a+b+c=\pi, a result old Euclid would agree with.
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