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 the class of third-order ODEs y'''=f(x‎, ‎y‎, ‎p‎, ‎q) as a Riemannian manifold‎ ‎where y'=p and y''=q‎, ‎if and only if q_{yy}=۰‎.‎Moreover‎, ‎we will see the linear second-order ODE with general form y''=\pm y+\beta(x) is the only case that is defined a minimal surface‎ ‎and is also totally geodesic‎.

Keywords:

Levi-Civita connection‎ , ‎minimal surface‎ , ‎moving frame‎ , ‎Riemannian manifold‎ , ‎Riemann curvature tensor‎ , ‎totally geodesic