# Metric and surfaces

1. Sep 16, 2014

### ChrisVer

Is it possible to find the parametric representation $X= x(y^{a})$ 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:
$E=1,~ F=0,~ G=\sin^2 u$ and $L=1, ~M=0,~N=\sin^2 u$ with $0 < u < \pi$
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 $κ_n = \frac{II}{I}$ is only a function of I and its derivatives.

Last edited: Sep 16, 2014