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

Abstract

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) as a Riemannian manifold where y ′ = p and y ′′ = q, if and only if qyy = 0. Moreover, we will see the linear second-order ODE with general form y ′′ = ±y + β(x) is the only case that is defined a minimal surface and is also totally geodesic.