According to the formula below
$$L = \int_{a}^{b}\sqrt{\Biggl|\sum_{i,j=1}^{n}g_{ij}(\gamma(t))\Bigl(\frac{\mathrm{d}}{\mathrm{d}t}x^{i}\circ\gamma(t)\Bigr)\Bigl(\frac{\mathrm{d}}{\mathrm{d}t}x^{j}\circ\gamma(t)\Bigr)\Biggr|}\mathrm{d}t,$$
one can compute the length of a curve $\gamma(t)$ between $t = a$ and $t=b$ having a metric $g_{ij}$. Under what necessary and/or sufficient one can do the inverse? I mean, having the length of the curve, interval $[a,b]$, and metric $g_{ij}$, how can we yield $\gamma(t)$?