Bonnet's type theorems in the relative differential geometry of the 4-dimensional space

We deal with hypersurfaces in the framework of the relative differential geometry in ${\mathbb{R}}^4$. We consider a hypersurface $\Phi$ in ${\mathbb{R}}^4$ with position vector field $\text{vect}x$ which is relatively normalized by a relative normalization $\text{vect}y$. Then $\text{vect}y$ is also a relative normalization of every member of the one-parameter family $\mathcal{F}$ of hypersurfaces ${\Phi }_{\mu }$ with position vector field $\text{vect}{x}_{\mu }=\text{vect}x+\mu \text{vect}y$, where $\mu$ is a real constant. We call every hypersurface ${\Phi }_{\mu }\in \mathcal{F}$ relatively parallel to $\Phi$. This consideration includes both Euclidean and Blaschke hypersurfaces of the affine differential geometry. In this paper we express the relative mean curvature’s functions of a hypersurface ${\Phi }_{\mu }$ relatively parallel to $\Phi$ by means of the ones of $\Phi$ and the “relative distance” $\mu$. Then we prove several Bonnet’s type theorems. More precisely, we show that if two relative mean curvature’s functions of $\Phi$ are constant, then there exists at least one relatively parallel hypersurface with a constant relative mean curvature’s function.