We study the high-frequency asymptotic analysis methods for the scattered fields by a cylindrically curved conducting surface excited by the incident wave on the curved surface from the convex side. We first derive the novel hybrid ray-mode solution for the scattered fields near the concave surface by solving a canonical problem formulated under the assumption that the cylindrically curved conducting surface possesses only one edge. Then by applying the ray tracing technique and the idea of Keller's GTD (Geometrical Theory of Diffraction), the solutions derived for the canonical problem are extended to account for the problem of the radiation from and the scattering by the other edge of the cylindrically curved surface. We confirm the validity of the novel asymptotic representations proposed in the present study by comparing both with the numerical results obtained from the method of moment and the experimental results performed in the anechoic chamber.