Given a metric $g_{\mu\nu}$ it is possible to find the equations of the geodesic on the Riemannian manifold $M$ defined by the metric itself:

$$\frac{d^2x^a}{ds^2} + \Gamma^{a}_{bc}\frac{dx^b}{ds}\frac{dx^c}{ds} = 0$$ where: $$\Gamma^a_{bc} = \frac{1}{2} g^{ad} \left( g_{cd,b} + g_{bd,c} - g_{bc,d} \right)$$ are the Christoffel symbols and $$g_{ab,c} = \frac{\partial {g_{ab}}}{\partial {x^c}}$$ Now, given a parametric equation of a curve, is it possible to find the metric of a Riemannian manifold which gives that curve as a geodesic? If the answer is 'Yes', is there a bijective correspondence between the curve and the metric? Or are there many metrics giving the same geodesic? Thanks in advance.