Is there a good way to guess for what indices the Christoffel symbols $\Gamma_{ij}^k$ vanish, for a Levi-Civita connection?

Question

Is there a good way to guess for what indices christoffel symbols, $\Gamma_{ij}^k$ vanish in general? For example, when calculating the Levi-Civita with spherical coordinates for a sphere most christoffel symbols vanish. What is the best way to guess which ones vanish in general?

Answer 1

You cannot say much in general, but you can say for certain classes of surfaces. For example surfaces of revolution have 3 zeros and 3 nonzero christoffel symbols. Also if a surface is flat then all christofferll symbols are zero.

Answer 2

In the frequently handled case where the metric is in diagonal form, that is, $g_{ij}=0$ if $i\neq j$, we have:

$$\begin{align} \Gamma^i_{\ jk} &= \frac{1}{g_{ii}}\Gamma_{ijk} &&\text{(no summation here!)} \\&= \frac{1}{2g_{ii}} \left(\partial_j g_{ik} + \partial_k g_{ij} - \partial_i g_{jk}\right) \end{align}$$ which implies that the Christoffel symbols (both first and second kind) are zero unless there are equal indices among $i,j,k$.

If, in addition to the metric being in diagonal form, the $g_{ii}$ are all independent from some coordinate (like longitude $\phi$), then if that coordinate index occurs exactly once among $i,j,k$, then it cannot index a nonzero metric coefficient, nor can it yield a nonzero partial derivative. This again results in the affected Christoffel symbols being zero.

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