Fourth Fundamental Form and $i$-th Curvature Formulas in ${\mathbb{E}}^{4}$

Abstract

We introduce fourth fundamental form $IV,$ and $i$-th curvature formulas of hypersurfaces in the four dimensional Euclidean geometry ${\mathbb{E}}^{4}$. Defining fourth fundamental form and $i$-th curvatures for hypersurfaces, we calculate them on rotational hypersurface. In addition we study rotational hypersurface satisfying $\Delta ^{IV}\mathbf{x=Ax}$ for some $4\times 4$ matrix $\mathbf{A.}$