The Geometry Of Submanifolds In Grassmannian Manifold

In this thesis, we investigate the harmonic two sphere in the Grassmann manifolds G(k, n), homogeneous three dimensional sphere in the complex projective space and surface with parallel mean curvature vector and Lagrangian surface in the hyperquadric Q2, by the method of moving frames. We construct a series of holomorphic differen-tial forms on S2, through which, we can simplify the moving frames along a harmonic two-sphere in G (2,4) and G (2,5). We construct many homogeneous three dimensional spheres in CPn by the irreducible unitary representations of SU(2), which are neither of CR-type nor weakly Lagrangian. We introduce the deviation angle8of surfaces with parallel mean curvature vector in Q2, and prove that there exists a series of isometric immersions with parallel mean curvature vector from a simply connected surface into Q2. Finally we describe a class of H-minimal Lagrangian surfaces with constant cur-vature in Q2, and give a example of minimal Lagrangian S2with Gaussian curvature K=2.