SUBMANIFOLDS OF EUCLIDEAN SPACES SATISFYING $\Delta H =AH$

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Published: Mar 1, 1995
DOI: https://doi.org/10.5556/j.tkjm.25.1994.4427
Keywords:
mean curvature vector ruled surfaces tubes

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BANG-YEN CHEN
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027, U.S.A.

Abstract




In [5] the author initiated the study of submanifolds whose mean curvature vector $H$ satisfying the condition $\Delta H =\lambda H$ for some constant $\lambda$ and proved that such submanifolds are either biharmonic or of 1-type or of null 2-type. Submanifolds of hyperbolic spaces and of de Sitter space-times satisfy this condition have been investigated and classified in [6,7]. In this article, we study submanifolds of $E^m$ whose mean curvature vector $H$ satisfies a more general condition; namely, $\Delta H =AH$ for some $m \times m$ matrix $A$.




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How to Cite
CHEN, B.-Y. (1995). SUBMANIFOLDS OF EUCLIDEAN SPACES SATISFYING $\Delta H =AH$. Tamkang Journal of Mathematics, 25(1), 71–81. https://doi.org/10.5556/j.tkjm.25.1994.4427
Issue
Vol. 25 No. 1 (1994)
Section
Papers
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References

C. Baikoussis and D. E. Blair, "On the Gauss map of rules surfaces," Glasgow Math. J., 34(1992), 355-359.

C. Baikoussis, B. Y. Chen and L. Verstraelen, "Ruled surfaces and tubes with finite type Gauss maps," Tokyo J. Math. 16(1993), 341-349.

B. Y. Chen, Total mean curvature and submanifolds of finite type, World Scientific, 1984.

B. Y. Chen, "Surfaces of finite type in Euclidean 3-space," Bull. Soc. Math. Belg. Sef B., 39(1987), 243-254.

B. Y. Chen, "Null 2-type surfaces in Euclidean space," Algebra, Analysis and Geomtry(1988), 1-18.

B. Y. Chen, "Some open problems and conjectures on submanifolds of finite type," Soochow J. Math., 17(1991), 169-188.

B. Y. Chen, "Some classification theorems for submanifolds in Minkowski space-time," Archiv Math., 62 (1994).

B. Y. Chen, "Submanifolds in de Sitter space-time satisfying $Delta H =lambda H$ ," preprint, 1992.

B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, "Ruled surface of finite type," Bull. Austral. Math. Soc., 42(1990), 447-453.

B. Y . Chen and M. Petrovic, "On spectral decomposition of immersions of finite type," Bull. Austral. Math. Soc., 44(1991), 117-129.

F. Dillen, J. Pas and L. Verstraelen, "On surfaces of finite type in Euclidean 3-space," Kodai Math. J., 13(1990). 10-21.

F. Dillen, J. Pas and L. Verstraelen, '.'On the Gauss map of surfaces of revolution," Bull. Inst. Math. Acad. Sinica, 18(1990), 239-246.

O. Garay, "An extension of Takahashi's theorem," Geom. Dedicata, 34(1930), 105-112.