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

TL;DR

This paper investigates the properties of relatively parallel hypersurfaces in n-dimensional relative differential geometry, focusing on the shape operator, curvatures, and affine normalization within a unified framework.

Contribution

It introduces a comprehensive analysis of the shape operator and curvature functions of relatively parallel hypersurfaces in n-dimensional relative differential geometry.

Findings

Derived expressions for the shape operator of relatively parallel hypersurfaces

Established relationships between relative principal curvatures and the shape operator

Analyzed the affine normalization of these hypersurfaces

Abstract

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