Communications of the
Korean Mathematical Society CKMS

The geodesics on the round 2-sphere $S^2$ are all simple closed curves of equal length. In 1903 Otto Zoll introduced other Riemannian surfaces with the same property. After that, his name is attached to the Riemannian manifolds whose geodesics are all simple closed curves of the same length. The question that ``whether or not the set of Zoll metrics on 2-sphere $S^2$ is connected?'' is still an outstanding open problem in the theory of Zoll manifolds. In the present paper, continuing the work of D.~Jane for the case of the Ricci flow, we show that a naive application of some famous geometric flows does not work to answer this problem. In fact, we identify an attribute of Zoll manifolds and prove that along the geometric flows this quantity no longer reflects a Zoll metric. At the end, we will establish an alternative proof of this fact.

Keywords: geometric flow, evolutionary equations, Zoll metrics, geodesic

MSC numbers: 53C44, 53C21, 53C22