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Torsion Tensor and Gauss Equation

  1. Jan 3, 2006 #1
    It's just something I am not sure and I can answer my question in another thread:
    In the Gauss Equation
    [tex]\partial^2_{i,j}(\vec{r})
    =\sum_{l=1}^m\Gamma^l_{i,j}\partial_l(\vec{r})+L_{i,j}\vec{n} [/tex]
    [tex]L_{i,j}=-\partial_i(\vec{n}) \partial_j(\vec{r})[/tex] has got something to do with the normal [tex]\vec{n}[/tex].
    So, when we use it to derive the Gauss-Codazzi Equation
    [tex]\partial_kL_{i,j}-\partial_j L_{i,k}
    =\sum_{l=1}^n (\Gamma^l_{i,k}L_{l,j}-\Gamma^l_{i,j}L_{l,k})[/tex]
    Should
    [tex]\sum_{l=1}^n (\Gamma^l_{i,k}L_{l,j}-\Gamma^l_{i,j}L_{l,k})[/tex]
    got something to do with the torsion tensor [tex]T(X,Y)=\nabla_X Y-\nabla_Y X[/tex] :confused:
     
    Last edited: Jan 3, 2006
  2. jcsd
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