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.



Length spectrum of spheres
Dear Igor,


The answer is that the length spectrum cannot determine the metric, not even the round metric : there are metrics on the sphere all of whose geodesics are closed and of the same length that are not isometric to the round metric. In fact, by a theorem of Guillemin (*The Radon transform on Zoll surfaces*. Advances in Mathematics 22 (1976), 85–119) there are lots and lots of Zoll surfaces.


If you pass to Finsler, even in the reversible case where there are lots of closed geodesics, there are more examples. For instance: take a three-dimensional normed space where the unit sphere $S$ is smooth and of positive Gaussian curvature (for any auxiliary Euclidean metric), then the length spectrum of $S$ equals the length spectrum of the sphere in the dual normed space (both with the induced length metric). This is in *Dual spheres have the same girth,* American journal of mathematics 128 (2), 361-371.


