  In this paper, the well-posedness of Cauchy's problem of fractional
Schr\"odinger equations with a power type nonlinearity on $n$-dimensional
manifolds with nonnegative Ricci curvature is studied. Under suitable volume
conditions, the local solution with initial data in $H^{[\frac{n}{2}]+1}$ will
blow up in finite time no matter how small the initial data is, which follows
from a new weight function and ODE inequalities. Moreover, the upper-bound of
the lifespan can be estimated.
