On spacelike Zoll surfaces with symmetries (Q267537)

In this paper, the authors present a thorough treatment of space-like Zoll surfaces with symmetries.NEWLINENEWLINENEWLINEMore precisely, a \textit{space-like Zoll surface} is a \(2\)-dimensional Lorentzian manifold all of whose space-like geodesics are closed, simple and have the same length. The basic example is the \(2\)-dimensional de Sitter space, i.e., the homogeneous Lorentzian manifold \({\mathrm{SO}}_0(2,1)/{\mathrm{SO}}_0(1,1)\) and its finite covers. Therefore, from the physical viewpoint, the spaces in the example can be regarded as highly symmetric solutions of the (\(2\)-dimensional hence trivial) Lorentzian vacuum Einstein equations with a positive cosmological constant. In an earlier work [Math. Z. 274, No. 1--2, 225--238 (2013; Zbl 1272.53016)], the authors exhibited a topological classification of these surfaces: all of them are diffeomorphic to either a cylinder or a Möbius strip. In this paper, they proceed further and address the problem of a finer classification up to conformal equivalence or even isometry. The main results of the paper are as follows:{\parindent=6mm \begin{itemize} \item[(i)] There exists an infinite-dimensional family of pairwise non-isometric space-like Zoll surfaces possessing a Killing field (see Theorems 5.6, 6.1, 7.5 in the paper);\item [(ii)] There exist Lorentzian Möbius strips of non-constant curvature all of whose space-like geodesics are closed (see Corollaries 5.10 and 7.9 in the paper). NEWLINENEWLINE\end{itemize}} This latter non-homogeneity result is interesting because the aforementioned de Sitter examples are homogeneous Lorentzian manifolds.