Hypersurfaces with $L_r$-pointwise $1$-type Gauss map

In this paper, we study hypersurfaces in $\mathbb E^{n+1}$ whose Gauss map $G$ satisfies the equation $L_rG = f(G + C)$ for a smooth function $f$ and a constant vector $C$, where $L_r$ is the linearized operator of the $(r + 1)$-st mean curvature of the hypersurface, i.e., $L_r(f)=\mathop{\mathrm{Tr}}(P_r\circ\nabla^2f)$ for $f\in \mathcal{C}^\infty(M)$, where $P_r$ is the $r$-th Newton transformation, $\nabla^2f$ is the Hessian of $f$, $L_rG=(L_rG_1,\ldots,L_rG_{n+1})$ and $G=(G_1,\ldots,G_{n+1})$. We focus on hypersurfaces with constant $(r+1)$-st mean curvature and constant mean curvature. We obtain some classification and characterization theorems for these classes of hypersurfaces.