Fourth fundamental form and i-th curvature formulas in \({{\mathbb {E}}}^{4}\)

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