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 $\varPhi$ 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 $\varPhi_\mu$ with position vector field $$\mathbf{x}_\mu = \mathbf{x} + \mu \, \mathbf{y},$$ where $\mu$ is a real constant. We call every hypersurface $\varPhi_\mu \in \mathcal{F}$ relatively parallel to $\varPhi$ 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 $\left( \varPhi_\mu,\mathbf{y}\right)$ to $\left(\varPhi,\mathbf{y}\right)$.