Topological and spectral rigidity of hypersurface Zoll manifolds

Abstract

We show that manifolds admitting Zoll families of minimal hypersurfaces are diffeomorphic to ${\mathbb{S}}^n$ or to $\mathbb{R}{\mathbb{P}}^n$, and we characterize their Zoll families in both cases. Then, we prove that a metric on $\mathbb{R}{\mathbb{P}}^3$ is surface Zoll if and only if its projective widths are all equal. Our last result asserts that a closed surface with the first four widths equal to $2\pi$ is isometric to $\mathbb{R}{\mathbb{P}}^2$ with its constant curvature one metric.

43 pages, 2 figures