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Metric and surfaces

  1. Sep 16, 2014 #1

    ChrisVer

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    Is it possible to find the parametric representation [itex]X= x(y^{a})[/itex] of a surface given the components of the metric alone?
    Reading the surface theory found in Differnetial Geometry of Martin M.Lipschutz I got the idea it's possible to do that given the metric components as well as the components of the 2nd Fundamental Form (through solving the Gauss-Weingarten formulas). In the book he does that for a 2sphere:
    [itex]E=1,~ F=0,~ G=\sin^2 u [/itex] and [itex]L=1, ~M=0,~N=\sin^2 u[/itex] with [itex] 0 < u < \pi[/itex]
    However in general I feel like the metric alone should be able to do the job. For example it also shows that the Gauss curvature [itex] κ_n = \frac{II}{I}[/itex] is only a function of I and its derivatives.
     
    Last edited: Sep 16, 2014
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