Abstract: A limiting case of the quantum mechanical scattering on the horned Riemannian manifolds is considered; namely, we assume the widths of the horns to be tending to zero. The scattering matrix is obtained in an explicit form, its unitarity is proved. Some examples are discussed in detail; in particular, in the case of Riemann surfaces of constant negative curvature a relation of the scattering amplitude to the Selberg zeta-function is explained. In the case of non-compact manifolds, an analogue of the optical theorem is presented.