Hypersurfaces in space forms satisfying the condition Δ𝑥=𝐴𝑥+𝐵

Alías, Luis J.; Ferrández, Angel; Lucas, Pascual

doi:10.1090/S0002-9947-1995-1257095-8

Hypersurfaces in space forms satisfying the condition $\Delta x=Ax+B$

Authors: Luis J. Alías, Angel Ferrández and Pascual Lucas
Journal: Trans. Amer. Math. Soc. 347 (1995), 1793-1801
MSC: Primary 53C40; Secondary 53C42, 53C50
DOI: https://doi.org/10.1090/S0002-9947-1995-1257095-8
MathSciNet review: 1257095
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Abstract: In this work we study and classify pseudo-Riemannian hypersurfaces in pseudo-Riemannian space forms which satisfy the condition $\Delta x = Ax + B$, where $A$ is an endomorphism, $B$ is a constant vector, and $x$ stands for the isometric immersion. We prove that the family of such hypersurfaces consists of open pieces of minimal hypersurfaces, totally umbilical hypersurfaces, products of two nonflat totally umbilical submanifolds, and a special class of quadratic hypersurfaces.

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