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 $T^1M$, 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 $T^1M / \mathbb{S}^1$, which arises as the quotient of the unit tangent bundle $T^1M$ by the $\mathbb{S}^1$-action induced by the geodesic flow.

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