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 \(\varPhi\) 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...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2017-10
Main Authors: Stamatakis, Stylianos, Kaffas, Ioannis
Format: Article
Language:English
Subjects:
Curvature
Differential geometry
Euclidean geometry
Fields (mathematics)
Geometry
Hyperspaces
Mathematical analysis
Theorems
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:2331-8422