By using the Sobolev inequality and Gradient estimates we have proved some rigidity theorems for the space-like submanifold in a de Sitter space to be totally umbilical under the global pinching conditions of second fundamental quantity on submanifolds.

By means of moving frame,some properties of ruled surfaces are obtained,including the surfaces being minimal,totally geodesic,totally developableand totally umbilical.

This paper studies totally real surfaces with parallel mean curvature vector in a 2 dimensional complex space form by use of f(x)= max u,v∈U xM‖B(u,u)-B(v,v)‖ 2,and obtains two Pinching theorems for totally umbilical submanifold.

In this paper,we study totally umbilical properties of complete submanifolds with parallel mean curvature vector in spheres,and generalize a concerned result of Alencar,do Carmo and Santos in the complete case.

If \$σ\-ξ<2n\$, then \$M\+n\$ is a totally umbilic submanifold.

In this paper through a thorough study of totally umbilical submanifolds in a recurrent space, we prove that a totally umbilical submanjfold in a recurrent space to be projective flat or concircular flat if and only if it is Einstein space (the dimension n≥4) which generalizes Z.

An important inequality on scalar curvatue and sectional curvature of minlocally symmetric space is obtained, as an application, some characteristics of umbilical submanifolds are obtained.

Making use of it, some characteristic of umbilical manifolds are obtained.

Let M be an umbilical Lorentzian isoparametric hypersurface of type II in the Lorentzian sphere Sn+11.

The invariant conformal property of umbilic point and totally umbilic surface in 3-dimensional compact Lorentz space Q3 are studied.

In this paper the classification of totally umbilic surfaces in three-dimensional Lorentz spaces R_1~3,S_1~3,H_1~3 is obtained first, next by the method of inbedding R_1~3,S_1~3,H_1~3 into their common comformal compactification Q~3, the classification of totally umbilic surfaces in Q~3 is obtained and the relation of those totally umbilic surfaces under the inbed is studied.

If SpecqM=Speq M , then M is also totally umbilical for dimension n<25.

In this paper, some properties of totally real and totally umbilical submanifolds of a complex projective space are obtained.

Some pinching theorems about totally real pseudo-umbilical submanifolds with parallel mean curvature vector becoming totallyreal and totally umbilical submanifolds are obtained by choosing a suitableframe ?eld.