• East Asian mathematical journal
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• v.14 no.2
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• pp.411-419
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• 1998

• Bulletin of the Korean Mathematical Society
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• v.31 no.2
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• pp.193-200
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• 1994

Let x : $M^{n}$ .rarw. $E^{m}$ be an isometric immersion of a manifold $M^{n}$ into the Euclidean space $E^{m}$ and .DELTA. the Laplacian of $M^{n}$ defined by -div.omicron.grad. The family of such immersions satisfying the condition .DELTA.x = .lambda.x, .lambda..mem.R, is characterized by a well known result ot Takahashi (8]): they are either minimal in $E^{m}$ or minimal in some Euclidean hypersphere. As a generalization of Takahashi's result, many authors ([3,6,7]) studied the hypersurfaces $M^{n}$ in $E^{n+1}$ satisfying .DELTA.x = Ax + b, where A is a square matrix and b is a vector in $E^{n+1}$, and they proved independently that such hypersurfaces are either minimal in $E^{n+1}$ or hyperspheres or spherical cylinders. Since .DELTA.x = -nH, the submanifolds mentioned above satisfy .DELTA.H = .lambda.H or .DELTA.H = AH, where H is the mean curvature vector field of M. And the family of hypersurfaces satisfying .DELTA.H = .lambda.H was explored for some cases in [4]. In this paper, we classify space curves x : R .rarw. $E^{3}$ satisfying .DELTA.x = Ax + b or .DELTA.H = AH, and find conditions for such curves to be equivalent.alent.alent.

• Bulletin of the Korean Mathematical Society
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• v.34 no.2
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• pp.155-171
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• 1997

Let $M^n$ be a connected subminifold of a Euclidean space $E^m$, equipped with the induced metric. Denoty by $\Delta$ the Laplacian operator of $M^n$ and by x the position vector. A well-known T. Takahashi's theorem [13] says that $\delta x = \lambda x$ for some constant $\lambda$ if and only if $M^n$ is either minimal subminifold of $E^m$ or minimal submanifold in a hypersphere of $E^m$. In [9], O. Garay studied the hypersurfaces $M^n$ in $E^{n+1}$ satisfying $\delta x = Dx$, where D is a diagonal matrix, and he classified such hypersurfaces. Garay's condition can be seen as a generalization of T.