The classical three-body problem is formulated as a problem of geodesic flows on a Riemannian manifold. It is proved that a curved space allows to detect new hidden symmetries of the internal motion of a dynamical system and reduces the three-body problem to the system of 6th order. It is shown that the equivalence of the original Newtonian three-body problem and the developed representation provides coordinate transformations together with an underdetermined system of algebraic equations. The latter makes the system of geodesic equations relative to the evolution parameter (internal time), i.e. to the arc length of the geodesic curve, irreversible.