# Torsion Tensor and Gauss Equation

1. Jan 3, 2006

### bchui

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

Last edited: Jan 3, 2006