Christoffel Symbols and Metric Tensor I know that the Christoffel Symbols are made out of the first derivative of Metric Tensor. Is there any relation between the number of metric components and Christoffel symbols? Is there any general rule that can tell me how many Christoffel symbols can be made out of the metric tensor?
For instance, we have only two diagonal metric components for the 2-sphere. We can only make two Christoffel symbols out of them. Is it a general rule?
 A: No, there is not such general rule that you are looking for. But you can count the maximum number of independent Christoffel symbols in a general space. In fact, for each independent component of the metric tensor, there are, at most, $N$ distinct Christoffel symbols.
Let me first start with an example. If you consider a two-dimensional Cartesian coordinate system as $$ds^2=dx^2+dy^2,$$
you cannot make any Christoffel symbols out of them, all of them are zero.
This counterexample shows that the metric of spacetime (flat or curved) which specifies the intrinsic geometry of space is very important. On the other hand, a certain space can be represented via different coordinate systems and, depending on the coordinate systems, we could have different Christoffel symbols. For example, consider the polar coordinate system, as a different representation of two-dimensional Cartesian coordinate system, by changing variables as
$$x = r\cos \phi,$$
$$y = r\sin \phi.$$
The resulting metric is now as
$$ds^2=dr^2+r^2d{\phi}^2,$$
but for the Christoffel symbols, one finds the following nonzero components
$$\Gamma _{\phi \phi }^r =  - r,\,\,\Gamma _{r\phi }^\phi  = \Gamma _{\phi r}^\phi  = \frac{1}{r}.$$
So, your answer is there is no such rule.
But there are some rules for general cases:
Consider a general $N$-dimensional space. For this space, there are, at most, $\frac{{N(N + 1)}}{2}$ independent components for the metric tensor. For each independent
component of the metric tensor, there are, at most, $N$ distinct Christoffel symbols. Therefore, the number of independent Christoffel symbols is obtained at most as
$$N \times \frac{{N(N + 1)}}{2} = \frac{{{N^2}(N + 1)}}{2}.$$
For example, for a general $2$-dimensional space, the total number of independent Christoffel symbols are, at most, $6$.
Now, consider an $N$-dimensional space with a diagonal metric. In this case there are $N$ independent components for the metric tensor. Therefore, The number of independent Christoffel symbols is obtained, at most, as
$$N \times N=N^2.$$
For example, for a general $2$-dimensional space with a diagonal metric, the total number of independent Christoffel symbols is, at most, $4$.
Finally, I think (not sure), depending on the isometries of a specific space, it is possible to count the total number of independent Christoffel symbols as well. For more information, see this paper.
