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Gauss Curvature

  1. Dec 19, 2007 #1
    I'm reviewing for my final and there is a question I cant seem to solve. If anyone could help me with it I would appreciate it very much.

    A ruled surface has the parameterization of the form:

    x(s,t) = A(s) + tB(s)

    where A(s) is unit speed, |B(s)| = 1.

    Show that: K<or= to 0.

    So, my first though was to just calculate the g_ij's and then just find its determinant and plug it into the equation:

    K = (R_1L21 * g_L2)/g ~ Summed for L = 1,2

    But after calculate some of the metric coeff's i'm not sure it will work out all that well. Any help would be appreciated.

    ' = d/ds

    g_12 = g_21 = <A',B>
    g_11 = 1 + 2t<A',B'> + t^2<B',B'>
    g_22 = 1
     
  2. jcsd
  3. Dec 22, 2007 #2
    See Do Carmo's "Differential Geometry of Curves and Surfaces" p.192.
     
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