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Media type:
E-Article
Title:
HYPERSURFACES IN SPACE FORMS SATISFYING THE CONDITION Lkx = Ax + b
Contributor:
Alías, Luis J.;
Kashani, S. M. B.
Published:
Mathematical Society of the Republic of China (Taiwan), 2010
Published in:
Taiwanese Journal of Mathematics, 14 (2010) 5, Seite 1957-1977
Language:
English
ISSN:
1027-5487;
2224-6851
Origination:
Footnote:
Description:
We study hypersurfaces either in the sphere Sn+1 or in the hyperbolic space ℍn+1 whose position vector x satisfies the condition Lkx = Ax + b, where Lk is the linearized operator of the (k + 1)-th mean curvature of the hypersurface for a fixed k = 0, . . . , n - 1, A ∊ ℝ(n+2)×(n+2) is a constant matrix and b ∊ ℝn+2 is a constant vector. For every k, we prove that when A is self-adjoint and b = 0, the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)-th mean curvature and constant k-th mean curvature, and open pieces of standard Riemannian products of the form ${S^m}\left( {\sqrt {1 - {r^2}} } \right) \times {S^{n - m}}\left( r \right) \subset {S^{n + 1}}$ with 0 r 1, and ${H^m}\left( {\sqrt {1 - {r^2}} } \right) \times {S^{n - m}}\left( r \right) \subset {H^{n + 1}}$ with r > 0. If Hk is constant, we also obtain a classification result for the case where b ≠ 0.