Equivalence of a harmonic oscillator to a free particle and Eisenhart lift

It is widely known in quantum mechanics that solutions of the Schrödinger equation (SE) for a linear potential are one-to-one correspondence with free SE. The physical reason this Einstein’s principle equivalence. What usually not so harmonic can also be mapped to equation. understanding equivalence as precisely case principle. We present geometric picture will link both above equivalences one constraint on Eisenhart metric.