# Christoffel symbols from definition or Lagrangian

1. Oct 19, 2007

### noospace

I asked this question in the tensor analysis formum but did we did not reach a satisfactory conclusion.

Here is the problem:

Let $\mathbf{x} : U \subset\mathbb{R}^2 \to S$ be a local parametrization of a regular surface S. Then the coefficients of the second derivatives of x in the basis of the tangent space are defined to be the Christoffel symbols.

On the other hand, if we write the first fundamental form $ds^2 = E du^2 + 2F du dv + G dv^2$ in differential form we have an extremization problem of the arc-length

Then the coefficients of of the corresponding Euler-Lagrange equations are essentially the Christoffel symbols.

Are there any interesting examples where the Lagrangian method of computing Christoffel symbols breaks down?