$\delta (2)$-ideal Euclidean hypersurfaces of null $L_1$-2-type

Abstract

We say that an isometric immersion hypersurface $ x:M^n\rightarrow\mathbb{E}^{n+1}$  is ofnull $L_k$-2-type  if  $x =x_1+x_2$, $ x_1, x_2:M^n\rightarrow\mathbb{E}^{n+1}$ are smooth maps and $L_k x_1 =0, ~ L_k x_2 =\lambda x_2$,  $\lambda$ is non-zero real number,  $L_k$ is the linearized operator ofthe $(k + 1)$th mean curvature of the hypersurface, i.e., $L_k( f ) =\text{tr} (P_k \circ \text{Hessian} f )$ for$f \in C^\infty(M)$, where $P_k$ is the $k$th Newton transformation,  $L_k x = (L_k  x_1, \ldots , L_k x_{n+1}), ~x = (x_1, \ldots, x_{n+1})$. In this article,  we classify $\delta (2)$-idealEuclidean hypersurfaces of  null $L_1$-2-type.