Complex submanifolds with splitting tangent sequences in rational homogeneous spaces of Picard number one and some related problems

Abstract

In the thesis we prove that a compact submanifold S ⊂ M with splitting tangent sequence is rational homogeneous when M is in a large class of rational homogeneous spaces of Picard number one. Moreover, when M is irreducible Hermitian symmetric, we prove that S must be also Hermitian symmetric. Motivated by the method proving the Hermitian symmetry (related to the restriction and projection of global vector fields), we give a differential geometric proof for the classification of compact splitting submanifolds with dimension ≥ 2 in a hyperquadric, which has been proven in [Jah05]using algebraic geometry. And using the same method we give a classification of compact splitting submanifolds with dimension ≥ 2 in low dimensional Grassmannian G(2, 3) as a new example. The new proof on classification of compact splitting submanifolds with dimension ≥ 2 in a hyperquadric provides motivation to study a weaker gap rigidity problem for Hermitian symmetric spaces of compact type. Roughly speaking we want to recover some standard models from their tangent spaces under certain assumptions. The case for diagonal curves in an irreducible compact Hermitian symmetric space of tube type is proven as a prototype which is also a dual analogy to a theorem obtained in [Mok02], and we generalize this to higher dimensional submanifolds in compact Hermitian symmetric spaces motivated by the proof of curve case.