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I found this equation

$$\sum_{k,l,r,s} g_{ik}\frac{\partial g_{jl}}{\partial x^{r}}g^{rk}g^{sl}\frac{\partial}{\partial x^{s}}=2\sum_{s}\Gamma_{ij}^{s}\frac{\partial}{\partial x^{s}}$$

here g's are coefficients of Riemannian metric and $\Gamma$'s are Christoffel symbols.

So, since $$2\sum_{l}\Gamma^{l}_{ij}g_{lk}=\frac{\partial g_{jk}}{\partial x^{i}} + \frac{\partial g_{ik}}{\partial x^{j}}-\frac{\partial g_{ij}}{\partial x^{k}}$$ how the first equation is possible, what happened with minus part?

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Those equations become a lot more readable in display style. You can get displayed equations by enclosing them in double dollar signs. There's an edit link underneath the question. –  joriki Oct 4 '12 at 14:23
    
@joriki I edited, because I couldn't read the question without doing so. Your general advice still stands, of course. –  Willie Wong Oct 4 '12 at 14:25
1  
In the first expression, you have a summation over $k$ of $\sum_k g_{ik} g^{kr}$, this becomes $\delta_i^r$. So in fact your first expression can be simplified to $$ \sum_{l} \partial_i g_{jl} g^{sl} = 2 \Gamma^s_{ij} $$ which does seem problematic. Where did you find this equation stated? –  Willie Wong Oct 4 '12 at 14:31
    
Here I found it: –  Novak Djokovic Oct 4 '12 at 14:48
    
H. Shima, The Geometry of Hessian Structures, page 24 –  Novak Djokovic Oct 4 '12 at 14:49

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