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 $y^{\prime \prime \prime }=f(x,y,p,q)$ as a Riemannian manifold where $y^{\prime }=p$ and $y^{\prime \prime }=q$, if and only if $q_{yy}=0$. Moreover, we will see the linear second-order ODE with general form $y^{\prime \prime }=\pm y+\beta (x)$ is the only case that is defined a minimal surface and is also totally geodesic.

Accepted for publication in Int. J. Nonlinear Anal. Appl