# Taylor expansion of Christoffel symbols

I would like to show that in geodesic coordinates with origin $p$

$$\Gamma_i = -\frac 12 \sum_j \mathrm{R}_p(\partial_i, \partial_j) x^j,$$

where $\Gamma_i = \Gamma_{ij}^k \partial_k \otimes \mathrm{d}x^j$ and $\mathrm{R}_p$ is the Riemann tensor at $p$. I tried some calculations, but I didn't get anywhere.

Is it true that $\nabla_j \Gamma_i = -\frac12 \mathrm{R}(\partial_i, \partial_j)$? Then the result would follow...

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