On the Tchebychev Vector Field in the Relative Differential Geometry

TL;DR

This paper explores the properties of hypersurfaces in Euclidean space through the decomposition of the Tchebychev vector field associated with relative normalizations, linking it to curvature and support functions.

Contribution

It introduces a novel decomposition of the Tchebychev vector field in relative differential geometry, enhancing understanding of hypersurface properties.

Findings

Decomposition of Tchebychev vector into two components

Relations between Tchebychev vector and Gaussian curvature

Insights into support function and hypersurface geometry

Abstract

In this paper we deal with relative normalizations of hypersurfaces in the (n+1)-dimensional Euclidean space $\mathbb{R}^{n+1}$. Considering a relative normalization $\bar{y}$ of an hypersurface $\Phi$ we decompose the corresponding Tchebychev vector $\bar{T}$ in two components, one parallel to the Tchebychev vector $\bar{T}_{EUK}$ of the Euclidean normalization $\bar{\xi}$ and one parallel to the orthogonal projection $\bar{y}_{T}$ of $\bar{y}$ in the tangent hyperplane of $\Phi$. We use this decomposition to investigate some properties of $\Phi$, which concern its Gaussian curvature, the support function, the Tchebychev vector field etc.