On Classifications Of Some Special Hypersurfaces

Geometry of submanifolds is an important topic of differential geometry,which is motived by a great number of geometers and topologists.Hypersurface,as a special kind of submanifolds,has a very wide range of research value.Especially,the classifications of hypersurfaces attract more and more scholars.In this paper,we study hypersurfaces in real space forms and Lorentz space forms.We also consider spacelike hypersurfaces in Lorentz warped product space and Riemannian warped product space.The main contents of this paper are as follows:Firstly,we study a class of special hypersurfaces in real space forms.In 2013,M.Cras-mareanu and C.E.Hretcanu[1,2]introduced the notion of golden struture on a manifold as a special tensor field of(1,1)-type on a manifold M and defined the golden-and product-shaped hypersurfaces in real space forms[3].Inspired by their work,we give the definition of generalized golden structure on a manifold and define the generalized golden-and product-shaped hypersur-faces.What is more,we obtain the classifications of the generalized golden-and product-shaped hypersurfaces in real space forms based on the classification of isoparametric hypersurfaces.Secondly,we consider a class of special hypersurfaces in Lorentz space forms.In 2014,D.Yang and Y.Fu[4]defined the golden-shaped hypersurface in Lorentz space forms and the whole families of the golden-shaped hypersurface in Lorentz space forms were obtained.Inspired by their work,we define the generalized golden-shaped hypersurface and give the classification of the generalized golden-shaped hypersurfaces in Lorentz space forms.Finally,the spacelike hypersurfaces immersed in Lorentz warped product space Mn+1=-I×fMn or Riemannian warped product space Mn+1=I×f Mn are studied.When the sectional curvature of fiber Mn is bounded from below,by applying the well known Omori-Yau maximum and minimum principles,under suitable restrictions on the mean curvature and the norm of the gradient of the height function,we obtain the uniqueness of spacelike hypersurfaces immersed in Lorentz warped product space and Riemannian warped product space.