Geometrical Structures Of The Product Manifolds S~3×S~3 And Its Submanifolds

Submanifold geometry is a very important research direction in differential geometry and 6 dimensional nearly Kaehler manifold is a hot research subject in the field of submanifold geometry.In 2005,Butruille developed an entire classification for 6 dimensional homognenous nearly Kaehler manifold,it must take one of the 6 dimensional sphere S6,product manifold S~3×S~3,complex project space CP3 and flat manifold SU(3)/U(1)×U(1)as its isomorphism.In this dissertation,we deeply study submanifolds of product manifold S~3×S~3 with nearly Kaehler structure and product structure.The main achievements of this dissertation are as follows:In Chapter 2,a characteristic description of angle function about Lagrangian submanifolds in nearly Kaehler manifolds S~3×S~3 is given.Firstly,nearly Kaehler structure J and almost product structure P are defined in nearly Kaehler S~3 ×S~3.Then,based on the classification of Lagrangian submanifolds,a Lagrangian immersion f=(pq,q)is found,which has the same induced metric with f=(p,q)and their angle functions satisfy ?i= 4/3?-?i,(i = 1,2,3).In Chapter 3,an integral inequality of 3 dimensional invariant submanifolds of the product manifold S~3×S~3 is constructed.According to the boundness of the squared length of second fundamental form,a Simons' type integral inequality of 3 dimensional invariant manifolds is developed,from which we conclude that the compact orientable minimal 3 dimensional submanifolds with parallel second fundamental and the compact orientable totally geodesic 3 dimensional submanifolds of S~3×S~3 are invariant submanifolds.In Chapter 4,the geometric relationships of nearly Kaehler product manifold M = M1×M2 and its F-invariant submanifolds N = N1×N2 are studied.First of all,we prove N1,N2 are totally geodesic submanifolds of N.Further,N is totally geodesic,minimal,pseudo umbilical,totally umbilical and curvature-invariant sub-manifold of M if and only if N1 and N2 are,respectively,totally geodesic,minimal,pseudo umbilical,totally umbilical and curvature-invariant submanifolds of M1 and M2.