We consider hypersurfaces in the real Euclidean space R n+1 (n ≥ 2) which are relatively normalized. We give necessary and sufficient conditions a) for a surface of negative Gaussian curvature in R 3 to be ruled, b) for a hypersurface of positive Gaussian curvature in R n+1 to be a hyperquadric and c) for a relative normalization to be constantly proportional to the equiaffine normalization.
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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