2-Type Hypersurfaces Satisfying $ \langle \Delta x , x-x_0 \rangle = \mbox{const.}$

Abstract

Let M be a connected n-dimensional submanifold of a Eu- clidean space En+k equipped with the induced metric and ∆ its Lapla- cian. If the position vector x of M is decomposed as a sum of three vectors x = x1 + x2 + x0 where two vectors x1 and x2 are non-constant eigen- vectors of the Laplacian , i.e., ∆xi = λixi,i = 1,2 ( λi ∈ R) and x0 is a constant vector, then, M is called a 2-type submanifold. In this paper we proved that a connected 2-type hypersurface M in En+1 whose postion vector x satisfies ⟨∆x, x − x0⟩ = c for a constant c, where ⟨ , ⟩ is the usual inner product in En+1, is of null 2-type and has constant mean curvature and scalar curvature.