Title: 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 $\left(\varPhi,\mathbf{y}\right)$.