Consider hyperbolic metrics on $\Sigma_g$ a closed orientable surface of genus $g$. Let $[\gamma_1] , \cdots, [\gamma_n]$ be a finite collection of isotopy classes of simple closed curves on $\Sigma_g$.

At what conditions on $[\gamma_1] , \cdots, [\gamma_n]$ is any (isotopy class of) metric completely determined by the lengths of the geodesic representatives of respective classes $[\gamma_1] , \cdots, [\gamma_n]$ ?

I am not necessarily looking for a complete answer and partial answers/comments are very welcome.