Zoll manifolds with boundary

Title: Zoll manifolds with boundary

Speaker: Eduardo Longa Rosinato

Abstract:

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.