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

TL;DR

This paper extends two classical Bonnet theorems to the context of 3-dimensional relative differential geometry, focusing on relatively parallel surfaces in Euclidean space and their properties.

Contribution

It formulates and proves the relative analogues of Bonnet's theorems for parallel surfaces within the framework of relative differential geometry.

Findings

Established relative Bonnet theorems for relatively parallel surfaces.

Derived conditions for the invariance of geometric properties under relative parallelism.

Extended classical differential geometry results to the relative setting.

Abstract

This paper is devoted to the 3-dimensional relative differential geometry of surfaces. In the Euclidean space $\R{E} ^3 $ we consider a surface $\varPhi %\colon \vect{x} = \vect{x}(u^1,u^2) $ with position vector field $\vect{x}$, which is relatively normalized by a relative normalization $\vect{y}% (u^1,u^2) $. A surface $\varPhi^*% \colon \vect{x}^* = \vect{x}^*(u^1,u^2) $ with position vector field $\vect{x}^* = \vect{x} + \mu \, \vect{y}$, where $\mu$ is a real constant, is called a relatively parallel surface to $\varPhi$. Then $\vect{y}$ 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~\cite{oB1853}).