Rigidity And Classification Of The Canonical Submanifolds In 6-dimensional Nearly K(?)hler Manifolds

6-dimensional nearly K(?)hler manifolds are an important class of geometric objects. It is very natural and important to study their various canonical submanifolds. In this doctoral dissertation, we study the almost complex surfaces and the Lagrangian submanifolds in 6-dimensional nearly K(?)hler manifolds, and obtain the following main results:(1) We derive a rigidity theorem for the compact almost complex surfaces in the homogeneous nearly K(?)hler S3× S3. We establish an integral inequality of Simons’ type for such surfaces, in which the equality holds if and only if the almost complex surfaces are totally geodesic. This fact gives a new characterization of the only two totally geodesic almost complex surfaces in S3× S3(see Theorem 1.1). Our integral inequality also provides a generalization of Simons’ integral inequality in the case that the ambient space is not locally symmetric and that the co-dimension of surfaces is 4.(2) We obtain a complete classification of the Lagrangian submanifolds in the homogeneous nearly K(?)hler S3× S3 with parallel second fundamental form. We first extend a result of the homogeneous nearly K(?)hler S6 to a general 6-dimensional strict nearly K(?)hler manifold, i.e. we prove that Lagrangian submanifolds in a6-dimensional strict nearly K(?)hler manifold with parallel second fundamental form must be totally geodesic(see Theorem 1.2). Moreover, by studying the almost product structure of S3× S3, we classify all the totally geodesic Lagrangian submanifolds in the nearly K(?)hler S3× S3(see Theorem 1.3).(3) We prove that isotropic Lagrangian submanifolds of 6-dimensional strict nearly K(?)hler manifolds are totally geodesic(see Theorem 1.4). This generalizes a result about the constant isotropic Lagrangian submanifolds of K(?)hler manifolds.We completely classify the J-isotropic Lagrangian submanifolds in the homogeneous nearly K(?)hler S3× S3(see Theorem 1.5). This classification theorem shows that by use of the condition of J-isotropic, we can give a unified characterization of all the totally geodesic Lagrangian submanifolds and those of constant sectional curvature in S3× S3.