and the second fundamental form is L = N = 0 , M = sin ⁡ ⁡ φ φ and the Gauss–Codazzi equation is φ φ u v = sin ⁡ ⁡ φ φ . Thus, any pseudospherical surface gives rise to a solution of the sine-Gordon equation, although with some caveats: if the surface is complete, it is necessarily singular due to the Hilbert embedding theorem. In the simplest case, the pseudosphere, also known as the tractroid, corresponds to a static one-soliton, but the tractroid has a singular cusp at its equator.