Let $M_1,M_2$ be two surfaces with the corresponding metrics $(g_{ij}),(h_{ij})$ if they both have the same Christoffel coefficients does $(g_{ij})=(h_{ij})$?
Intuitively the Christoffel coefficients are expression of the metric of the surface so we may find two different metrics with the same Christoffel coefficients?
Let $\lambda >0$ and $(M,g)$ be a Riemannian manifold. Then for the inverse of the metric matrix of $\lambda g$ we have $(\lambda g)^{kl}=\lambda^{-1}g^{kl}$. It follows that the Christoffel symbols with respect to $(M,\lambda g)$ are given by
$\Delta^k_{ij} = \frac{1}{2}(\lambda g)^{kl}((\lambda g)_{kl,i}-(\lambda g)_{ij,k} + (\lambda g)_{li,j}) = \frac{1}{2} \lambda^{-1}g^{kl}(\lambda g_{kl,i}-\lambda g_{ij,k}+\lambda g_{li,j}) = \Gamma^k_{ij} $
where $\Gamma^k_{ij}$ are the Christoffel symbols of $(M,g)$. However the Christoffel symbols determine a system of inhomogenous, non-linear, first-order PDE with indeterminants being the entries $g_{ij}$ from which one can in principle derive a space of solutions in which the metric $g$ lies.
