The Ribaucour transformation generalises the notion of parallel surfaces and Darboux transformations. This paper introduces the construction of Ribaucour transforms via the Cauchy-Kovalevskaya theorem. The paper gives an explicit integral formula of Ribaucour transformations for surfaces of revolution. The paper also studies the singularities appearing on Ribaucour transforms and gives the criteria for cuspidal edges, swallowtails, cuspidal cross caps, and \(D_4^\pm\) singularities.
In Section 2, Theorem 1 states that any Ribaucour transformation can be obtained by solving the damped wave equation with the help of the Cauchy-Kovalevskaya theorem. Theorem 2 provides an explicit integral formula for Ribaucour transformations for surfaces of revolution, which is used to create examples in later sections. The examples demonstrate the ability to construct non-linear Weingarten and non-isothermic Ribaucour transformations from surfaces of revolution. The Ribaucour transformations can be applied locally to any regular surface immersed in a space form, specifically rotational surfaces, without imposing any other geometric property on the transformed surface. In previous articles, the transformations were required to preserve the geometric property of being linear Weingarten (which includes surfaces of constant Gaussian or mean curvature), however in this paper the Ribaucour transformations are applied to rotational surfaces in \(\mathbb{R}^3\), without imposing any other geometric property on the transformed surface.
In Section 3, the paper analyses singularities on Ribaucour transformations and establishes criteria for cuspidal edges, swallowtails, cuspidal cross caps, and \(D_4^\pm\) singularities. Parallel surfaces are noted as typical examples of Ribaucour transformations and the results of previous studies on singularities on parallel surfaces are generalized to Ribaucour transformations and some geometrical analysis is given.