Extension of two Bonnet’s theorems to the 3-dimensional relative differential geometry

Abstract

This paper is devoted to the 3-dimensional relative differential geometry of surfaces. In the Euclidean space $$\mathbb {E} ^3 $$ we consider a surface $$\varPhi $$ with position vector field $$\varvec{x}$$ , which is relatively normalized by a relative normalization $$\varvec{y} $$ . A surface $$\varPhi ^* $$ with position vector field $$\varvec{x}^* = \varvec{x} + \mu \, \varvec{y}$$ , where $$\mu $$ is a real constant, is called a relatively parallel surface to $$\varPhi $$ . Then $$\varvec{y}$$ is also a relative normalization of $$\varPhi ^*$$ . 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 Bonnet in Nouv Ann de Math 12:433–438, 1853).