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

2017/01/29 by Stamatakis, Stylianos, Kaffas, Ioannis, Delivos, Ioannis
#53A05 #53A15 #53A40 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1701.09086

Abstract

This paper is devoted to the 3-dimensional relative differential geometry of surfaces. In the Euclidean space \RE ³ we consider a surface \varPhi %\colon \vectx = \vectx(u¹,u²) with position vector field \vectx, which is relatively normalized by a relative normalization \vecty% (u¹,u²) . A surface \varPhi^*% \colon \vectx^* = \vectx^*(u¹,u²) with position vector field \vectx^* = \vectx + μ \vecty, where μ is a real constant, is called a relatively parallel surface to \varPhi. Then \vecty 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~\citeoB1853).

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