Line congruence and transformation of projective surfaces

Abstract

The aim of this article is to present and reformulate systematically what is known about surfaces in the projective 3-space, in view of transformations of surfaces, and to complement this with some new results. Special emphasis will be laid on line congruences and Laplace transformations. A line congruence can be regarded as a transformation connecting one focal surface with the other focal surface. A Laplace transformation is regarded as a method of constructing a new surface from a given surface by relying on the asymptotic system the surface is endowed with. A principal object in this article is a class of projectively minimal surfaces. We clarify the procedure of getting new projectively minimal surfaces from a given one, which was found by F. Marcus, as well as the procedure of Demoulin transformation of projective surfaces.