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

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This paper is devoted to the 3-dimensional relative differential geometry of surfaces. In the Euclidean space E ³ we consider a surface % x = x(u¹,u²) with position vector field x, which is relatively normalized by a relative normalization y% (u¹,u²). A surface % x* = x*(u¹,u²) with position vector field x* = x + μ y, where μ is a real constant, is called a relatively parallel surface to. Then y is also a relative normalization of. 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 oB1853).