Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Metric and surfaces

  1. Sep 16, 2014 #1


    User Avatar
    Gold Member

    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
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted