# Levi-Civita Connection and vanishing christoffel symbols

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?

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.

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.

• So,apart from the plane is there no case when number of zeroes > 3 ? – Narasimham Mar 29 '16 at 19:59
• I am sure there is, I just cannot remember it from the top of my head. – user26977 Mar 29 '16 at 20:01
• Please send me a comment then. – Narasimham Mar 29 '16 at 20:07
• Your statement that all the Christoffel symbols vanish for a flat surface is wrong. It does hold , however, when the first fundamental form coefficients are constants. – Ted Shifrin Mar 29 '16 at 20:17