For producing a cylindrical surface that has a surface structure of predetermined geometry suitable for application of material by thermal spraying, a geometrically predetermined groove structure of minimal depth and width is introduced into the surface by a tool embodied as a follow-on tool in that a groove cross-section is processed successively to a final size. In order for the surface to be producible in mass production with constant quality, the groove structure is worked in such that first a base groove is introduced with a groove bottom width that is smaller than the groove bottom width of the finished groove. Subsequently, at least one flank of the base groove is processed for producing an undercut groove profile by a non-cutting action or cutting action wherein the introduced groove structure is deformed in such a way that the groove openings are constricted by upsetting deformations of material.