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

Abstract

We deal with hypersurfaces in the framework of the $n$-dimensional relative differential geometry. We consider a hypersurface $\Phi$ of ${\mathbb{R}}^{n+1}$ with position vector field $\mathbf{x}$, which is relatively normalized by a relative normalization $\mathbf{y}$. Then $\mathbf{y}$ is also a relative normalization of every member of the one-parameter family $\mathcal{F}$ of hypersurfaces ${\Phi }_{\mu }$ with position vector field $${\mathbf{x}}_{\mu }=\mathbf{x}+\mu \mathbf{y},$$ where $\mu$ is a real constant. We call every hypersurface ${\Phi }_{\mu }\in \mathcal{F}$ relatively parallel to $\Phi$ at the "relative distance" $\mu$. 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 $\left({\Phi }_{\mu },\mathbf{y}\right)$ to $\left(\Phi ,\mathbf{y}\right)$.

13 pages. arXiv admin note: text overlap with arXiv:1707.07549