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On the Tchebychev Vector Field in the Relative Differential Geometry

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 $Φ$ we decompose the corresponding Tchebychev vector $\bar{T}$ in two components, one parallel to the Tchebychev vector $\bar{T}_{EUK}$ of the Euclidean normalization $\barξ$ and one parallel to the orthogonal projection $\bar{y}_{T}$ of $\bar{y}$ in the tangent hyperplane of $Φ$. We use this decomposition to investigate some properties of $Φ$, which concern its Gaussian curvature, the support function, the Tchebychev vector field etc.