Topological and spectral rigidity of hypersurface Zoll manifolds

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 $2π$ is isometric to $\mathbb{R}\mathbb{P}^2$ with its constant curvature one metric.