On Problems Related To The Submanifolds Of Spherical Immersion

The submanifolds of spherical immersion is very important research objects in submanifold geometry,which is widely studied by many geometers.In this dissertation,we study the classification problem of the equivariant minimal 3-dimensional sphere S~3 of the complex projective space CP~n,the rigidity problem for compact Lagrangian submanifolds in the homogeneous nearly K?hler S~6(the characterization of Berger sphere)and Legendrian submanifolds in Sasakian space forms(the characterization of contact Whitney sphere),respectively.The main results are as follows.(1)We study the equivariant CR minimal immersion from S~3into CP~n.Without the assumption of constant sectional curvatures,by employing the equivariant condition which implies that the induced metric is left-invariant and that all geometric properties of S~3endowed with a left-invariant metric can be expressed in terms of the structure constants of the Lie algebra su(2).Thus,we establish a classification theorem for equivariant CR minimal immersions from the 3-sphere S~3into CP~n(see Theorem 1.4).(2)We study the equivariant minimal immersion from S~3into CP~n.Under the additional condition(?_xF)(X=0,(X?ker(see(1.1.4)),we have completely classified such kind of submanifolds(see Theorem 1.7).Our classification theorem covers the result of Theorem 1.5.Moreover,we also construct a typical example of equivariant non-minimal immersion from S~3into CP~n satisfying(?_xF)X=0,which is neither Lagrangian nor of CR type.(3)We obtain the rigidity theorem for compact Lagrangian submanifolds in the nearly Kahler S~6.We establish a Simons'type integral inequality for compact Lagrangian submanifolds of S~6(1).Moreover,we show that the equality sign occurs if and only if the Lagrangian submanifold is either the totally geodesic S~3(1)or the Dillen-Verstraelen-Vrancken's Berger sphere S~3(see Theorem 1.14).This fact gives a new characterization of two Lagrangian sphere in nearly Kahler S~6.(4)We obtain the rigidity theorem for Legendrian submanifolds in Sasakian space forms.We establish a pointwise inequality for Legendrian submanifolds of any Sasakian space form with mean curvature vector and scalar curvature,and classify all Legendrian submanifolds which realize the equality case of the inequality(see Theorem 6.21).As the more in-depth results,we establish a general inequality dealing with Legendrian submanifolds in the Sasakian space form,which involves the squared of norm of the covariant differentiation of both the second fundamental form and the mean curvature vector field,and classify all Legendrian submanifolds which realize the equality case of the inequality(see Theorem 1.17).Finally,we give a characterization of the contact Whitney spheres in all Sasakian space forms.