[en] There is given a revision of the formulation and the proof of the theorem regarding the global unique solvability in the class of weak (energy) solutions of the Cauchy problem, for a second-order semilinear pseudodifferential hyperbolic equation on a smooth Riemannian manifold (of dimension n ≥ 3) without boundary. Under natural additional assumptions it is proved that if the initial data u(0, x), ∂_tu(0, x) are smoother: u(0)element-of H^S+1, ∂_tu(0)element-of H⁵, 0 < S ≤ 2, then also the weak solution is smoother: u element-of C([0,τ]→H^S+1), ∂_tu element-of C([0,τ]→H^S). 26 refs., 2 tabs