Studies On Hypersurfaces As Ricci Solitons In Lorentz Space

In this paper,we study hypersurfaces in Lorentz space E1n+1.We discuss the case that shape operators are diagonalizable and not diagonalizable respectively.We give the conditions that hypersurfaces are Ricci solitons whose potential vector fields are the tangential part of their position vectors.In addition,the classification of the Ricci solitons are obtained.The main results of this paper are as follows:1.The hypersurface M with diagonalizable shape operator in Lorentz space E1n+1 is a Ricci soliton whose potential vector field is the tangent part of its position vector,then M has one simple constant curvature or has two distinct principle curvatures;Under the assumption that the hypersurface M has constant mean curvature,such Ricci soliton is locally isometry to the hyperbolic space Hn(-c2)or de Sitter space S1n(c2)or Hn-1(-c2)× E or S1n-1(c2)×E.2.A hypersurface with an undiagonalized shape operators and constant mean curvature in Lorentz space E1n+1 is a Ricci soliton whose tangential part of its position vector is the potential vector field if and only if the principal curvatures of the hypersurface are all equal and the shape operator A is of the form iii);(See page 19.)Under the assumption that the hypersurfaces M has constant mean curvature,such Ricci solitons are locally isometry to the generalized umbilical hypersurfaces given by equation(3.10).