Christoffel symbols

Remarks

They determine $\nabla$ on $U$: $\nabla_X Y = \sum_k \big( X Y^k + \sum_{i,j} \Gamma^k_{ij} X^i Y^j \big) \partial_k$ for $X = \sum X^i \partial_i$ and $Y = \sum Y^j \partial_j$. For the Levi-Civita connection of a Riemannian metric with components $g_{ij}$ and inverse matrix $(g^{kl})$, $$\Gamma^k_{ij} = \tfrac12 \sum_l g^{kl} \big( \partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij} \big).$$ Example: for the flat connection of $\R^2$ in polar coordinates $(r, \theta)$, where $g = dr^2 + r^2 d\theta^2$, this formula gives $\Gamma^r_{\theta\theta} = -r$, $\Gamma^\theta_{r\theta} = \Gamma^\theta_{\theta r} = 1/r$, and all other symbols vanish. Non-example: the $\Gamma^k_{ij}$ are not the components of a tensor field; under a change of chart they transform with an extra term containing second derivatives of the coordinate change. So they can vanish at a point in one chart and not in another; in Cartesian coordinates the symbols above are all $0$. The connection is torsion-free exactly when $\Gamma^k_{ij} = \Gamma^k_{ji}$ in every chart.