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Thread: 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 $\displaystyle a, b, c$.
    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 $\displaystyle \pi-a+\pi-b+\pi - c=3\pi-a-b-c$. So we end up with $\displaystyle 2\pi-(3\pi-a-b-c)=0$ or $\displaystyle a+b+c=\pi$, a result old Euclid would agree with.
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