I have 2 dimensional Riemannian manifold with Levi-Civita connection defined as follows:
$\Gamma^1_{jk}=0$ and $\Gamma^2_{jk}(x^1,x^2)$ - any functions. I try to characterize metrics that are compatible with these Christoffel symbols. Therefore I used metric compatibility condition: $\nabla g=0$ and result with 8 PDEs:
$\frac{\partial g_{ab}}{\partial x^c}-\Gamma^d_{ac}g_{db}-\Gamma^d_{bc}g_{ad}=0$
However I am not able to solve them. Are they even solvable? Should I use different method?
I will be gratefull for any sugestions. Best regards.