Plane curves and geodesic diffeomorphisms (Q5956164)

In a pseudo-Riemannian space \((M, g)\), the author defines an \(r\)-plane curve as a curve whose \(r\)-th curvature vanishes. He proves that every diffeomorphism transferring \(r\)-plane curves of \((M, g)\) into \(r\)-plane curves of another pseudo-Riemannian space \((\overline{M}, \overline{g})\) (for a fixed \(r\)) is a geodesic diffeomorphism, i.e., it transfers geodesics into geodesics.