As all the information about the spacetime structure is being contained in the metric, it should be possible to express the Christoffel symbols in terms of this metric.
The following calculation is a little bit long and requires special attention (although it is not particularly difficult).
So far, we have defined both the metric tensor and the Christoffel symbols as respectively:
Let's begin by rewriting our metric tensor in the slightly different form gαμ:
Now, in this second step, we want to calculate the partial derivative of gαμ by xν:
Now let's try to rewrite the Christoffel symbol by multiplying each part of the equation by the partial derivative of ξσ relative to xβ:
We can now rewrite the partial derivative of gαμ by xν as follows:
or we recognize from our previous article Generalisation of the metric tensor that
If we now do the operation (3) + (4) - (5) we get:
Finally the last step consists in multiplying both sides of the equations by the inverse metric gβα to isolate the Christoffel symbol
Usually, we adopt the following convention for writing partial derivatives: