Is there a good way to guess for what indices the Christoffel symbols $\Gamma_{ij}^k$ vanish, for a Levi-Civita connection? 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? 
 A: 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.
You might also be interested in the thread concerning Is there a good way to compute Christoffel Symbols.
A: 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.
