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

TL;DR

This paper explores the relative differential geometry of hypersurfaces in 4D space, deriving Bonnet-type theorems that relate constant relative mean curvatures of hypersurfaces and their relatively parallel counterparts.

Contribution

It introduces Bonnet's type theorems in the context of relative differential geometry of hypersurfaces in 4D, linking constant mean curvature functions to relatively parallel hypersurfaces.

Findings

Expressed relative mean curvature functions in terms of original hypersurface and relative distance

Proved existence of relatively parallel hypersurfaces with constant mean curvature functions

Extended classical Euclidean and affine differential geometry results to 4D relative setting

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.