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 $\mathbb{R}{E}^3$ we consider a surface $\Phi$ with position vector field $\vect{x}$, which is relatively normalized by a relative normalization $\vect{y}% (u^1,u^2) $. A surface ${\Phi }^{\ast }$ with position vector field $\vect{x}^* = \vect{x} + μ\, \vect{y}$, where $\mu$ is a real constant, is called a relatively parallel surface to $\Phi$. Then $\vect{y}$ is also a relative normalization of ${\Phi }^{\ast }$. 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~\cite{oB1853}).

arXiv comment9 pages