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A simple counterexample is the space $\mathbb R^n$, with a metric $g_{ab}$ independent on the point. As examples, the Euclidean space $\mathbb R^n$, but also the Minkowski spacetime. The geodesics are the lines in $\mathbb R^n$, no matter how we choose the constant metric $g_{ab}$. Hence, different metrics can give the same set of geodesics.