Is it possible to find the parametric representation [itex]X= x(y^{a})[/itex] of a surface given the components of the metric alone?(adsbygoogle = window.adsbygoogle || []).push({});

Reading the surface theory found inDiffernetial GeometryofMartin M.LipschutzI 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.

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# Metric and surfaces

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