Document Type : Research Paper

Authors

• Zeynab Bakhshandeh-Chamazkoti ¹
• Abolfazl Behzadi ¹
• Rohollah Bakhshandeh-Chamazkoti ²
• Mehdi Rafie-Rad ¹

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 $y^{\prime }=p$ 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

• Levi-Civita connection‎
• ‎minimal surface‎
• ‎moving frame‎
• ‎Riemannian manifold‎
• ‎Riemann curvature tensor‎
• ‎totally geodesic