Zoll Manifolds and Complex Surfaces

Abstract

We classify compact surfaces with torsion-free affine connections for which every geodesic is a simple closed curve. In the process, we obtain completely new proofs of all the major results [4] concerning the Riemannian case. In contrast to previous work, our approach is twistor-theoretic, and depends fundamentally on the fact that, up to biholomorphism, there is only one complex structure on CP 2 .

Smooth Projective Planes

“…Theorem 3 (Salzmann [44,42], [43] 51.29, Löwen [36], LeBrun & Mason [35]). Two-dimensional smooth projective planes are diffeomorphic to the real projective plane.…”

Self-dual Zollfrei conformal structures withα-surface foliation

“…We call a projective structure for a projectively equivalent class [∇]. The following proposition is prooved in [6] 2. L and d are integrable.…”

“…LeBrun and L. J. Mason investigated two kinds of twistor-type correspondences in [6] and [7]. One of them is the correspondence for the Zoll projective structure on two-dimensional manifolds ( [6]). A projective structure is an equivalence class of torsion-free connections under the projective equivalence, where two torsion-free connections are called projectively equivalent if they have exactly the same unparameterized geodesics.…”

Singular self-dual Zollfrei metrics and twistor correspondence

“…An example of such a metric is the standard metric on S 2 . A Zoll projective structure on M is a projectively equivalence class of torsion-free connections on the tangent bundle T M whose geodesics are all closed, where two torsion-free connections are said to be projectively equivalent if and only if they have exactly the same unparameterized geodesics ( [10]). Each Zoll metric defines a Zoll projective structure.…”