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

Abstract

We deal with hypersurfaces in the framework of the relative differential geometry in $\mathbb{R}^4$. We consider a hypersurface $\varPhi$ in $\mathbb{R}^4$ with position vector field $\vect{x}$ which is relatively normalized by a relative normalization $\vect{y}$. Then $\vect{y}$ is also a relative normalization of every member of the one-parameter family $\mathcal{F}$ of hypersurfaces $\varPhi_\mu$ with position vector field $\vect{x}_\mu = \vect{x} + \mu \, \vect{y}$, where $\mu$ is a real constant. We call every hypersurface $\varPhi_\mu \in \mathcal{F}$ relatively parallel to $\varPhi$. 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 $\varPhi_\mu$ relatively parallel to $\varPhi$ by means of the ones of $\varPhi$ 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 $\varPhi$ are constant, then there exists at least one relatively parallel hypersurface with a constant relative mean curvature's function.