Length spectrum rigidity and flexibility of spheres of revolution with one equator

TL;DR

This paper investigates the length spectrum rigidity of spheres of revolution with one equator, showing that under certain symmetries, the spectrum uniquely determines the metric and describing the structure of isospectral classes.

Contribution

It introduces a notion of marked length spectrum for symmetric metrics on the sphere and proves spectral rigidity results, including uniqueness and classification of isospectral metrics.

Findings

Isospectral metrics have conjugate geodesic flows.

Under $ ext{Z}_2$-symmetry, the spectrum determines the metric.

Each isospectral class contains a unique symmetric metric, forming an infinite-dimensional convex set.

Abstract

We define a notion of marked length spectrum for $S^1$-symmetric Riemannian metrics on the two-sphere having only one equator. We prove that isospectral metrics in this class have conjugate geodesic flows. Under a further $\mathbb{Z}_2$-symmetry assumption, we show that the marked length spectrum determines the metric. Finally, we show that every isospectral class of metrics contains a unique $\mathbb{Z}_2$-symmetric metric and give an explicit description of this isospectral class as an infinite dimensional convex set, generalizing the known description of $S^1$-symmetric Zoll metrics. This paper contains also two appendices, in which we provide an elementary proof of the fact that a $C^2$ real valued function on an interval is determined by the set of tangent lines to its graph, and we classify a class of $S^1$-invariant contact forms on three-manifolds.