Generalization of two Bonnet's Theorems to the relative Differential Geometry of the 3-dimensional Euclidean space

Abstract

This paper is devoted to the 3-dimensional relative differential geometry of surfaces. In the Euclidean space E 3 we consider a surface Φ with position vector field x, which is relatively normalized by a relative normalization y. A surface Φ * with position vector field x * = x + µ y, where µ is a real constant, is called a relatively parallel surface to Φ. Then y is also a relative normalization of Φ * . The aim of this paper is to formulate and prove the relative analogues of two well known theorems of O. Bonnet which concern the parallel surfaces (see [1]).