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.

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