  Let $M$ be a complex manifold. We prove that a compact submanifold $S\subset
M$ 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. The basic tool we use is the
restriction and projection map $\pi$ 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 $\dim\geq 2$ in a hyperquadric, which has been previously
proven using algebraic geometry.
