Splitting submanifolds in rational homogeneous spaces of Picard number one

Let M be a complex manifold. We prove that a compact submanifold S ⊂ M with splitting tangent sequence (called a splitting submanifold) 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. These cover some of the results given in [Jah05]. The basic tool we use is the restriction and projection map π of the global holomorphic vector fields on the ambient space which is induced from the splitting condition. The usage of global holomorphic vector fields may help us set up a new scheme to classify the splitting submanifolds in explicit examples, as an example we give a differential geometric proof for the classification of compact splitting submanifolds with dim ≥ 2 in a hyperquadric, which has been proven in [Jah05] using algebraic geometry.