Geometry of submanifolds of all classes of third-order ODEs as a Riemannian manifold

In this paper, we prove that any surface corresponding to linear second-order ODEs as a submanifold is minimal in all classes of third-order ODEs $y’‘’=f(x, y, p, q)$asaRiemannianmanifoldwhere$y’=p$and$y’‘=q$,ifandonlyif$q_{yy}=0$\mathrm{.}Moreover,wewillseethelinearsecond-orderODEwithgeneralform$y’'=\pm y+\beta(x)$ is the only case that is defined a minimal surface and is also totally geodesic.