  In this paper, we establish two new types of invariant sets for the coupled
nonlinear Schrodinger system on $\mathbb{R}^n$, and derive two sharp thresholds
of blow-up and global existence for its solution. Some analogous results for
the nonlinear Schrodinger system posed on the hyperbolic space $\mathbb{H}^n$
and on the standard 2-sphere $\mathbb{S}^2$ are also presented. Our arguments
and constructions are improvements of some previous works on this direction. At
the end, we give some heuristic analysis about the strong instability of the
solitary waves.
