On the shape operator of relatively parallel hypersurfaces in the n-dimensional relative differential geometry

الأصل · EN

We deal with hypersurfaces in the framework of the n-dimensional relative differential geometry. We consider a hypersurface of Rⁿ⁺¹ with position vector field x, which is relatively normalized by a relative normalization y. Then y is also a relative normalization of every member of the one-parameter family F of hypersurfaces μ with position vector field xμ= x + μ y, where μ is a real constant. We call every hypersurface μ∈ F relatively parallel to at the "relative distance" μ. In this paper we study (a) the shape (or Weingarten) operator, (b) the relative principal curvatures, (c) the relative mean curvature functions and (d) the affine normalization of a relatively parallel hypersurface (μ,y) to (,y).