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Christoffel symbols from definition or Lagrangian

  1. Oct 19, 2007 #1
    I asked this question in the tensor analysis formum but did we did not reach a satisfactory conclusion.

    Here is the problem:

    Let [itex]\mathbf{x} : U \subset\mathbb{R}^2 \to S[/itex] 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 [itex]ds^2 = E du^2 + 2F du dv + G dv^2[/itex] 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?
  2. jcsd
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