The Umbilicity Of Linear Weingarten Submanifolds In Pseudo-Riemannian Space F Orms

Pseudo-Riemannian manifolds are a class of differential manifolds with pseudo-Riemannian metrics,which are a generalization of Riemannian manifolds.In particular,a complete and simply connected pseudo-Riemannian space with constant curvature is called a pseudo-Riemannian space form.This paper mainly studies the rigidity problems of a class of submanifolds in pseudo-Riemannian space form Nqn+p(c)with arbitrary index.The second fundamental form of space-like submanifold is time-like,space-like,or lightlike in pseudo-Riemannian space form with arbitrary index.In this regard,this paper mainly focuses on the case that the second fundamental form is locally time-like.By using the methods of moving frame and tensor analysis,we systematically study the rigidity problems of linear Weingarten submanifolds.The specific works are as follows:Firstly,constructing a suitable Cheng-Yau modifiled operator L on submanifold,and we apply L to the square norm of the traceless tensor ? to obtain an inequality about ?.Applying the generalized maximum principle to the inequality,we find the total umbilical condition of this submanifold,that is,when the supremum of the traceless tensor is less than a constant ?,it is a total umbilical submanifold.Especially,when the supremum of traceless tensor is equal to the constant ?,this submanifold is isometric to a product manifold H1(c-ccoth2 r)× Sn-1(c-ctanh2 r)(c>0),Rn-1 × H1(-coth2 r)(c=0),or Hn-1(c+ctan2 r)× H1(c+ccot2 r)(c<0).Secondly,considering the umbilicity of a complete linear Weingarten space-like submanifold with parallel normalized mean curvature vector immersed in pseudo-Riemannian space form,we prove the submanifold with nonnegative curvature is either totally umbilical submanifold or isometric to Mk,k=1,...,r,where the factors Mk,mutually perpendicular along their intersections,are totally umbilical submanifolds of Nqn+p(c).Finally,we use some algebraic inequalities to estimate equality and prove that when the square norm of the second fundamental form of the linear Weingarten submanifold with parallel normalized mean curvature vector in Sqn+p(c)is less than(?).this submanifold is a totally umbilical submanifold.When ?=(?),the submanifold is isometric to a product manifold Sn-1(c-ctanh2 r)× H1(c-ccoth2 r).This paper mainly classifies a class of linear Weingarten space-like submanifolds in pseudo-Riemannian space forms according to the geometric quantities of submanifolds.It generalizes the classical known results of submanifolds with constant scalar curvatures,and also provides more in-depth theoretical support for the study of the rigidity of submanifold.