Geometric properties of the self-adjoint Sturm-Liouville problems

In this paper, geometric properties of the self-adjoint Sturm-Liouville problems are investigated. It is proved that for linear self-adjoint Sturm-Liouville problems, the eigenfunctions correspond exactly to the projections of the curvature lines on the energy functional surface with an appropriate metric and that the eigenvalues correspond exactly to the principal curvatures (at the origin) of the same energy functional surface.