Say I have a metric with components $g_{ij}$ in some specified coordinate chart covering a domain $\Omega$:
$$ds^2 = g_{ij} dx^i dx^j.$$
I can, as is standard, compute the Christoffel symbols in this coordinate chart by just taking appropriate combinations of derivatives of the metric components:
$$\Gamma^i_{jk} = \frac{1}{2} \sum_n g^{in} \left(\partial_j g_{nk} + \partial_k g_{nj} - \partial_n g_{jk}\right).$$
My question is the following: if two metrics $g^{(1)}$ and $g^{(2)}$ have the same Christoffel symbols in this chart, must their components $g^{(1)}_{ij}$ and $g^{(2)}_{ij}$ in this chart also agree (perhaps under the assumption that $g^{(1)}_{ij} = g^{(2)}_{ij}$ at the boundary of $\Omega$)? One might naively suppose that they should, since the metric-compatibility condition $\nabla_i g_{jk} = 0$ gives an over-determined system of linear first-order PDEs for the components:
$$0 = \nabla_i g_{jk} = \partial_i g_{jk} - \sum_n \left(\Gamma^n_{ij} g_{nk} + \Gamma^n_{ik} g_{jn} \right).$$
Then I might expect that upon fixing boundary conditions at the boundary of $\Omega$, I should be able to find a unique solution to the above PDEs given a specified set of $\Gamma^i_{jk}$. However, I realize that uniqueness of solutions to PDEs is never as obvious as one would like, so I'm wondering whether this argument is too simplistic.