On the Tchebychev Vector Field in the Relative Differential 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.