Complex Geometric Invariants Associated to Zoll Manifolds.

Abstract

In this thesis, we explore several invariants associated to Zoll manifolds such as the manifold of geodesics and the ways to embed them into complex spaces. We study the construction by Lebrun and Mason on Zoll surfaces and the Grauert tube constructions in the CROSSes. We prove that the only Zoll spheres with an infinite tube are the round ones. Using a similar argument, we prove a special case of Burns' algebraicization conjecture. In the study of the Lebrun-Mason construction, we obtain some new ideas about the embedded holomorphic disks on the quadric in CP^3.