The purpose of the present paper is to introduce a notion of geodesic flows— simple integrability. In a word, a simply integrable geodesic flow is a geodesic flow which can be integrated by a single quadratic function. A remarkable property of simple integrability is the duality: To a Riemannian manifold with simply integrable geodesic flow, there corresponds, through certain conformal change of the metric, such another Riemannian manifold. To be more precise, let (M, g) be an n-dimensional Riemannian manifold (n≧2)
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