I am new to general relativity and I am currently facing an apparent inconsistency in the definition of the connection coefficients.
Some references I've been consulting (e.g. the lecture notes by S. Carroll or those by D. Tong) define them as follows:
$$
\nabla_{\partial_i}\partial_j=\Gamma^k_{ij}\partial_k
$$
where $\{\partial_i\}$ are the chart-induced basis vector fields.
Some other references, however (e.g. these beautiful lectures by prof. Schuller) define them with the lower indices swapped:
$$
\nabla_{\partial_i}\partial_j=\Gamma^k_{ji}\partial_k
$$
I know that for a torsion-free connection the coefficients are symmetric in the lower indices, so it doesn't really matter which definition one chooses to adopt. Still, this disagreement is bothering me, since I would like to know how the coefficients are supposed to be defined in the general case of a non-torsion-free connection. Is this discrepancy due to a mere convention/choice of notation? And if not, which definition is to be assumed as the correct one?
Best Answer
Let me start by echoing what Deane said in his comment: it's just a matter of convention, and you can choose whichever convention you prefer. Most differential geometers use the first convention you mention, because it seems more natural to have $i$ and $j$ in the same order on both sides of the equation.
But there's also a reasonable justification for the other convention. In many applications of connections, it's important to consider the matrix $\theta^k_j$ of connection forms with respect to a particular local frame $(E_j)$, which are $1$-forms defined by $$ \nabla_X E_j = \sum_k \theta^k_j(X) E_k. $$ For example, a connection is compatible with a metric if and only if the connection forms with respect to an orthonormal frame are skew-symmetric (i.e., take their values in the Lie algebra $\mathfrak{o}(n)$).
If you use the second convention for the Christoffel symbols, then the relation between the connection forms and the Christoffel symbols is $$ \theta^k_j = \sum_i \Gamma^k_{ji} \varepsilon ^i, $$ where $(\varepsilon^i)$ is the dual coframe to $(E_j)$. From this point of view, the second convention for the Christoffel symbols seems a little more natural. I didn't watch enough of Schuller's lecture to see if this is where he was headed, but my guess is that this is why he chose the convention he did.