Bulletin of the
Korean Mathematical Society BKMS

We say that an isometric immersed hypersurface $x:M^n\to\mathbb R^{n+1}$ is of $L_k$-finite type ($L_k$-f.t.) if $x=\sum_{i=0}^px_i$ for some positive integer $p<\infty$, $x_i:M\to\mathbb R^{n+1}$ is smooth and $L_kx_i=\lambda_ix_i, \lambda_i\in\mathbb R,0\leq i\leq p$, $L_kf={\rm tr} P_k\circ\nabla^2f$ for $f\in C^\infty(M)$, where $P_k$ is the $k$th Newton transformation, $\nabla^2f$ 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 $S^m(r)\times\mathbb R^{n-m}$, ruled surfaces in $\mathbb R^{n+1}$ and some revolution surfaces in $\mathbb R^3$.

Keywords: hypersurfaces, ($L_1$-)finite type

MSC numbers: 53A05, 53B25, 53C40