In this paper, we investigate rotational hypersurfaces family in $n$ -dimensional Euclidean space ${ \mathbb{ E}}^{ n}$. Our focus is on studying the Gauss map $\mathcal{ G}$ of this family with respect to the operator $\mathbb{ L}_{ k}$, which acts on functions defined on the hypersurfaces. The operator $\mathbb{ L}_{ k}$ can be viewed as a modified Laplacian and is known by various names, including the Cheng--Yau operator in certain cases. Specifically, we focus on the scenario where $k=n-3$ and $n\geq 3$. By applying the operator $\mathbb{ L}_{ n-3}$ to the Gauss map $\mathcal{ G}$, we establish a classification theorem. This theorem establishes a connection between the $n\times n$ matrix $\mathcal{ A}$, and the Gauss map $\mathcal{ G}$ through the equation $\mathbb{ L}_{ n-3}\mathcal{ G}=\mathcal{ A}\mathcal{ G}$.
Categories: math.DG