Note on isometry invariant geodesic on two dimensional spherical manifold (Q2639354)

Given an isometry f on a Riemannian manifold M, any geodesic c(t) is called f-invariant if \(f(c(t))=c(t+1)\). Suppose for f that there exists a unique geodesic from every point x to f(x) and call it a small displacement of M. When M is homeomorphic to \(S^ 2\), it can be shown that if M admits a small displacement f, then there exists always an f- invariant geodesic which is closed.