ON SOME L1-FINITE TYPE (HYPER)SURFACES IN ℝn+1

Title & Authors
ON SOME L1-FINITE TYPE (HYPER)SURFACES IN ℝn+1

Abstract
We say that an isometric immersed hypersurface x : $\small{M^n\;{\rightarrow}\;{\mathbb{R}}^{n+1}}$ is of $\small{L_k}$-finite type ($\small{L_k}$-f.t.) if $\small{x\;=\;{\sum}^p_{i=0}x_i}$ for some positive integer p < $\small{\infty}$, $\small{x_i}$ : $\small{M{\rightarrow}{\mathbb{R}}^{n+1}}$ is smooth and $\small{L_kx_i={\lambda}_ix_i}$, $\small{{\lambda}_i\;{\in}\;{\mathbb{R}}}$, $\small{0{\leq}i{\leq}p}$, $\small{L_kf=trP_k\;{\circ}\;{\nabla}^2f}$ for $\small{f\;{\in}\$, where $\small{P_k}$ is the kth Newton transformation, $\small{{\nabla}^2f}$ is the Hessian of f, $\small{L_kx\;=\;(L_kx^1,\;{\ldots},\;L_kx^{n+1})}$, $\small{x=(x^1,\;{\ldots},\;x^{n+1})}$. In this article we study the following(hyper)surfaces in $\small{{\mathbb{R}}^{n+1}}$ from the view point of $\small{L_1}$-finiteness type: totally umbilic ones, generalized cylinders $\small{S^m(r){\times}{\mathbb{R}}^{n-m}}$, ruled surfaces in $\small{{\mathbb{R}}^{n+1}}$ and some revolution surfaces in $\small{{\mathbb{R}}^3}$.
Keywords
hypersurfaces;($\small{L_{1^-}}$)finite type;
Language
Korean
Cited by
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SURFACES IN $\mathbb{E}^3$ WITH L1-POINTWISE 1-TYPE GAUSS MAP, Bulletin of the Korean Mathematical Society, 2013, 50, 3, 935
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Some Integral Formulas for the (r+ 1)th Mean Curvature of a Closed Hypersurface, International Journal of Mathematics and Mathematical Sciences, 2012, 2012, 1
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On Some -Finite-Type Euclidean Hypersurfaces, ISRN Geometry, 2012, 2012, 1
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Quadric hypersurfaces of L r -finite type, Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2013, 54, 2, 625
6.
On some L 1-finite type Euclidean surfaces, Acta Mathematica Vietnamica, 2013, 38, 2, 303
7.
CLASSIFICATIONS OF HELICOIDAL SURFACES WITH L1-POINTWISE 1-TYPE GAUSS MAP, Bulletin of the Korean Mathematical Society, 2013, 50, 4, 1345
8.
HYPERSURFACES IN 𝕊4THAT ARE OF Lk-2-TYPE, Bulletin of the Korean Mathematical Society, 2016, 53, 3, 885
9.
Surfaces in $$\mathbb {S}^3$$ S 3 of $$L_1$$ L 1 -2-Type, Bulletin of the Malaysian Mathematical Sciences Society, 2016
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