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Paper   IPM / M / 127
School of Mathematics
  Title:   Hypersurfaces in IR1N satisfying ∆x=Ax+B
  Author(s):  S. M. B. Kashani
  Status:   Published
  Journal: Algebras Groups Geom.
  Vol.:  13
  Year:  1996
  Pages:   81-91
  Supported by:  IPM
In this paper we study hypersurfaces Msn in \BbbR1n+1 (or in \BbbRn+1) verifying the equation ∆x = Ax+B and the condition that the principal curvatures of the surface is not (−n ∈ ) times the mean curvature at the points where the mean curvature is nonzero.
    We prove that the mean curvature of the surface is constant and as a result, either Msn has zero mean curvature or when Msn has at most two different principal curvatures, it is isoparametric. Here Msn is a (pseudo) Riemannian manifold with metric of signature s,s=0,1,\BbbR1n+1 is the (n+1)-dimensional flat Lorentzian space, A is an endomorphism of \BbbR1n+1 and B ∈ \BbbR1n+1, ∆ is the Laplacian operator on Msn, x: Msn → \BbbR1n+1 is an isometric immersion.

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