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Since the theory of the second order ordinary differential equations is nontrivial, that is, there are equations which may not be mapped to each other by a change of dependent and independent variables, their differential geometric study is reasonable in the language of G-structures. The present talk is based on the study of Riemannian structure on a manifold associated to a second order ODE in appropriate jet bundle. We firstly define unique connection with values in the Lie algebra of the orthogonal group by means of the Pfaffian system encoding a differential equation and show that 2-graph (or 2-jet) of an integral curve of given equation describes a geodesic curve of this metric compatible connection. We mainly focus on the equations of simple, damped and forced harmonic oscillators since they lead to an interesting class of three dimensional Riemannian manifolds. Namely, Riemannian manifolds identified with these equations locally admit Lie group structures with (non-isomorphic) solvable Lie algebras. Also, any equation for these oscillatory motions describes a Riemannian manifold of non-positive constant scalar curvature depending on certain values of the oscillation frequency and the friction coefficient, which may be though as deformation parameters for underlying Riemannian structure