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arXiv · 2204.04926

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

Z. Bakhshandeh-Chamazkoti·A. Behzadi·R. Bakhshandeh-Chamazkoti·M. Rafie-Rad
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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.

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Z. Bakhshandeh-Chamazkoti, A. Behzadi, R. Bakhshandeh-Chamazkoti, M. Rafie-Rad. 2022-04-11. Geometry of submanifolds of all classes of third-order ODEs as a Riemannian manifold. https://doi.org/10.22075/ijnaa.2022.25069.2913

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