Topological and spectral rigidity of hypersurface Zoll manifolds

Abstract: We show that manifolds admitting Zoll families of minimal hypersurfaces are diffeomorphic to $\mathbb{S}<sup>{n}$ or to $\mathbb{R}\mathbb{P}<sup>{n}$, and we characterize their Zoll families in both cases. Then, we prove that a metric on $\mathbb{R}\mathbb{P}<sup>3$ 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 $2π$ is isometric to $\mathbb{R}\mathbb{P}<sup>2$ with its constant curvature one metric.