Geometry Seminar: Jakob Dittmer (U Münster)

Speaker: Jakob Dittmer (U Münster)

Title: The Berger conjecture for three-dimensional manifolds.

Abstract: A conjecture of Berger states that on a simply connected manifold M in which all geodesics are closed, all geodesics have the same length. In this thesis, we prove that this conjecture holds for three-dimensional manifolds. The proof is carried out in two parts.

In the first part, we use the Morse theory of the free loop space of M to deduce that the geodesic flow acts freely on the unit tangent bundle T¹M, or that it has at least two distinct singular orbit types.

In the second part, we apply methods from symplectic geometry and topology to investigate the topology of the symplectic orbifold T¹M / S¹, which arises as the quotient of the unit tangent bundle T¹M by the S¹-action induced by the geodesic flow.

Combining these two approaches ultimately yields a proof of Berger’s conjecture for three-dimensional manifolds.