We provide a shorter and more transparent proof of a result by I. Oleinik [25, 26, 27] . It gives a sucient condition of the essential self-adjointness of a Schr odinger operator on a non-compact Riemannian manifold with a locally bounded potential in terms of the completeness of the dynamics for a related classical system. The simplication of the proof given by I. Oleinik is achieved by an explicit use of the Lipschitz analysis on the Riemannian manifold and also by additional geometrization

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... guments which include a use of a metric which is conformal to the original one with a factor depending on the minorant of the potential.