It is well known, since the classical work of Darboux and Zoll, that the length spectrum of a Riemannian 2-sphere does not provide sufficient information to recover the Riemannian metric. In this talk, based on joint work with Alberto Abbondandolo, I will introduce a suitable notion of marked length spectrum on each S^1-symmetric Riemannian metric on the 2-sphere having only one equator, and study the rigidity and flexibility imposed by these data.
While the marked length spectrum does not recover the metric, I will show that it recovers at least the contact geometry of the unit tangent bundle: isospectral metrics have conjugate geodesic flows.
Under a further Z_2-symmetry assumption, the marked length spectrum does recover the metric. Indeed, every isospectral class of metrics contains a unique Z_2-symmetric metric, and I will give an explicit description of every isospectral class as an infinite-dimensional convex set, generalizing the known description of S^1-symmetric Zoll metrics.
