We say that an isometric immersed hypersurface x : $M^n;{ ightarrow};{mathbb{R}}^{n+1}$ is of $L_k$-finite type ($L_k$-f.t.) if $x;=;{sum}^p_{i=0}x_i$ for some positive integer p < $infty$, $x_i$ : $M{ ightarrow}{mathbb{R}}^{n+1}$ is smooth and $L_kx_i={lambda}_ix_i$, ${lambda}_i;{in};{mathbb{R}}$, $0{leq}i{leq}p$, $L_kf=trP_k;{circ};{ abla}^2f$ for $f;{in}'C^{infty}(M)$, where $P_k$ is the kth Newton transformation, ${ abla}^2f$ is the Hessian of f, $L_kx;=;(L_kx^1,;{ldots},;L_kx^{n+1})$, $x=(x^1,;{ldots},;x^{n+1})$. In this article we study the following(hyper)surfaces in ${mathbb{R}}^{n+1}$ from the view point of $L_1$-finiteness type: totally umbilic ones, generalized cylinders $S^m(r){ imes}{mathbb{R}}^{n-m}$, ruled surfaces in ${mathbb{R}}^{n+1}$ and some revolution surfaces in ${mathbb{R}}^3$.