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.
Bakhshandeh-Chamazkoti,Z , Behzadi,A , Bakhshandeh-Chamazkoti,R and Rafie-Rad,M . (2023). Geometry of submanifolds of all classes of third-order ODEs as a Riemannian manifold. International Journal of Nonlinear Analysis and Applications, 14(1), 1283-1294. doi: 10.22075/ijnaa.2022.25069.2913
MLA
Bakhshandeh-Chamazkoti,Z , , Behzadi,A , , Bakhshandeh-Chamazkoti,R , and Rafie-Rad,M . "Geometry of submanifolds of all classes of third-order ODEs as a Riemannian manifold", International Journal of Nonlinear Analysis and Applications, 14, 1, 2023, 1283-1294. doi: 10.22075/ijnaa.2022.25069.2913
HARVARD
Bakhshandeh-Chamazkoti Z, Behzadi A, Bakhshandeh-Chamazkoti R, Rafie-Rad M. (2023). 'Geometry of submanifolds of all classes of third-order ODEs as a Riemannian manifold', International Journal of Nonlinear Analysis and Applications, 14(1), pp. 1283-1294. doi: 10.22075/ijnaa.2022.25069.2913
CHICAGO
Z Bakhshandeh-Chamazkoti, A Behzadi, R Bakhshandeh-Chamazkoti and M Rafie-Rad, "Geometry of submanifolds of all classes of third-order ODEs as a Riemannian manifold," International Journal of Nonlinear Analysis and Applications, 14 1 (2023): 1283-1294, doi: 10.22075/ijnaa.2022.25069.2913
VANCOUVER
Bakhshandeh-Chamazkoti Z, Behzadi A, Bakhshandeh-Chamazkoti R, Rafie-Rad M. Geometry of submanifolds of all classes of third-order ODEs as a Riemannian manifold. IJNAA. 2023;14(1):1283-1294. doi: 10.22075/ijnaa.2022.25069.2913
International Journal of Nonlinear Analysis and Applications