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. & 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. https://doi.org/10.22075/ijnaa.2022.25069.2913
MLA
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", 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
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." 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