It is known (thanks to Hingston, Bangert, Franks, Birckhoff, etc) that $(S^2, g)$ has lots of primitive closed geodesics for any Riemannian metric $g$ (Riemannian is crucial here, this is not true for Finsler metrics).The question is: does the length spectrum determine the metric? This is obviously a completely different question from the similar question for hyperbolic surfaces, where spectral methods are available. In particular, it is not even obvious that if closed geodesics are the same length, then $g$ is the (appropriately scaled) round metric.