Semisurfaces and the equations of Codazzi-Mainardi

Abstract

In this paper, we shall describe the equations of Codazzi-Mainardi for a surface without any umbilical point, using the gradient vector fields of the mean curvature function and the Gaussian curvature function. In addition, based on this description, we shall obtain a homogeneous polynomial, which is an analogue of a Codazzi-Mainardi polynomial obtained in [3], and we shall find a relation between two polynomials.