  We prove global weighted Strichartz estimates for radial solutions of linear
Schr\"odinger equation on a class of rotationally symmetric noncompact
manifolds, generalizing the known results on hyperbolic and Damek-Ricci spaces.
This yields classical Strichartz estimates with a larger class of exponents
than in the Euclidian case and improvements for the scattering theory. The
manifolds, whose volume element grows polynomially or exponentially at
infinity, are characterized essentially by negativity conditions on the
curvature, which shows in particular that the rich algebraic structure of the
Hyperbolic and Damek-Ricci spaces is not the cause of the improved dispersive
properties of the equation. The proofs are based on known dispersive results
for the equation with potential on the Euclidean space, and on a new one, valid
for C^1 potentials decaying like 1/r^2 at infinity.
