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

TL;DR

This paper investigates the geometric structure of submanifolds associated with third-order ODEs, establishing conditions under which these surfaces are minimal or totally geodesic within a Riemannian framework.

Contribution

It characterizes when surfaces from second-order ODEs are minimal in third-order ODE classes, identifying specific forms that yield minimal or totally geodesic surfaces.

Findings

Surfaces from linear second-order ODEs are minimal iff q_{yy}=0.

The form y''=± y + β(x) uniquely yields minimal and totally geodesic surfaces.

Minimality is characterized by a specific second derivative condition.

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 $q_{yy}=0$. 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.