• East Asian mathematical journal
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• v.34 no.5
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• pp.643-649
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• 2018

Let M be a connected n-dimensional submanifold of a Euclidean space $E^{n+k}$ equipped with the induced metric and ${\Delta}$ its Laplacian. If the position vector x of M is decomposed as a sum of three vectors $x=x_1+x_2+x_0$ where two vectors $x_1$ and $x_2$ are non-constant eigenvectors of the Laplacian, i.e., ${\Delta}x_i={\lambda}_ix_i$, i = 1, 2 (${\lambda}_i{\in}R$) and $x_0$ is a constant vector, then, M is called a 2-type submanifold. In this paper we proved that a connected 2-type hypersurface M in $E^{n+1}$ whose postion vector x satisfies ${\langle}{\Delta}x,x-x_0{\rangle}=c$ for a constant c, where ${\langle}$, ${\rangle}$ is the usual inner product in $E^{n+1}$, is of null 2-type and has constant mean curvature and scalar curvature.

• Journal of the Korean Mathematical Society
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• v.54 no.4
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• pp.1163-1173
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• 2017

In this paper, we introduce the condition (I) (cf. Section 2) and prove that there is no nontrivial tangential holomorphic vector field of a certain hypersurface of infinite type in ${\mathbb{C}}^2$.