  In this paper, we partially settle down the long standing open problem of the
finite time blow-up property about the nonlinear Schr$\ddot{o}$dinger equations
on some Riemannian manifolds like the standard 2-sphere $S^2$ and the
hyperbolic 2-space $H^{2}(-1)$. Using the similar idea, we establish such
blow-up results on higher dimensional standard sphere and hyperbolic $n$-space.
Extensions to $n$-dimensional Riemannian warped product manifolds with $n\geq
2$ are also given.
