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

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

2 Department of Mathematics, Faculty of Basic Sciences, Babol Noshirvani University of Technology, Babol, Iran

Abstract

‎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 $q_{yy}=0$‎.
‎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.

Keywords

Volume 14, Issue 1
January 2023
Pages 1283-1294
  • Receive Date: 31 October 2021
  • Revise Date: 21 January 2022
  • Accept Date: 26 January 2022