Zoll manifolds are Riemannian manifolds all of whose geodesics are closed and have the same length. Beyond the round sphere, nontrivial examples were constructed by Funk and Guillemin, initiating a rich line of research. In this talk, I introduce a free-boundary analogue of this notion. A compact Riemannian manifold with boundary is said to be Zoll with boundary if every geodesic issuing orthogonally from the boundary returns orthogonally and is nowhere tangent to it. I will show that such manifolds exhibit strong rigidity: all free-boundary geodesics have the same length and share the same Morse index. Using Morse index theory and algebraic topology, we obtain a complete geometric and topological classification. In particular, when the boundary is connected, the manifold is a tubular neighborhood of a closed embedded submanifold (the “soul”), and the boundary fibers over the soul either as a sphere bundle or as a nontrivial two-fold covering. This is joint work with Paolo Piccione and Roney Santos.

The eigenvalues of the Laplace-Beltrami operator on a closed Riemannian manifold are very natural geometric invariants. Although in many problems the Riemannian structure is kept fixed, the eigenvalues can be seen as functionals in the space of metrics. This is the suitable setting for the calculus of variations. In this vein, El Soufi and Ilias have characterised the metrics which are critical for the first eigenvalue among all metrics of fixed volume and among all metrics of fixed volume in a conformal class. In the talk, I will prove a similar characterisation for some critical metrics which are induced by embeddings into a fixed Riemannian manifold.