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We deal with hypersurfaces in the framework of the $n$-dimensional relativedifferential geometry. We consider a hypersurface $\varPhi$ of$\mathbb{R}^{n+1}$ with position vector field $\mathbf{x}$, which is relativelynormalized by a relative normalization $\mathbf{y}$. Then $\mathbf{y}$ is alsoa 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 realconstant. We call every hypersurface $\varPhi_\mu \in \mathcal{F}$ relativelyparallel 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 $\left(\varPhi,\mathbf{y}\right)$.

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