We consider relative normalizations of ruled surfaces with non-vanishing
Gaussian curvature $K$ in the Euclidean space $\mathbb{R} ^{3}$, which are
characterized by the support functions $^{\left( \alpha \right) }q=\left \vert
K\right \vert ^{\alpha}$ for $\alpha \in \mathbb{R}$ (Manhart's relative
normalizations). All ruled surfaces for which the relative normals, the Pick
invariant or the Tchebychev vector field have some specific properties are
determined. We conclude the paper by the study of the affine normal image of a
non-conoidal ruled surface.Comment: 12 page
This paper is devoted to the 3-dimensional relative differential geometry of surfaces. In the Euclidean space E 3 we consider a surface Φ with position vector field x, which is relatively normalized by a relative normalization y. A surface Φ * with position vector field x * = x + µ y, where µ is a real constant, is called a relatively parallel surface to Φ. Then y is also a relative normalization of Φ * . 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 [1]).
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