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

TL;DR

This paper introduces the concepts of the fourth fundamental form and $i$-th curvature formulas for hypersurfaces in four-dimensional Euclidean space, providing calculations for rotational hypersurfaces and exploring their properties under specific differential operators.

Contribution

It defines the fourth fundamental form and $i$-th curvatures in ${ m I ext{-}E}^4$, and computes these for rotational hypersurfaces, extending classical differential geometry concepts.

Findings

Defined the fourth fundamental form $IV$ for hypersurfaces in ${ m E}^4$

Derived formulas for $i$-th curvatures in ${ m E}^4$

Analyzed rotational hypersurfaces satisfying $ riangle^{IV}oldsymbol{x}=oldsymbol{A}oldsymbol{x}$

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 $\mathbf{A.}$