  As is well known, the exact evolution equation of the curvature perturbation
plays a very important role in investigation of the inflation power spectrum of
the flat universe. However, its corresponding exact extension for the non-flat
universes has not yet been given out clearly. The interest in the non-flat,
specially closed, universes is being aroused recently. The need of this
extension is pressing. We start with most elementary physical consideration and
obtain finally this exact evolution equation of the curvature perturbation for
the non-flat universes, as well as the evolutionary controlling parameter and
the exact expression of the variable mass in this equation. We approximately do
a primitive and immature analysis on the power spectrum of non-flat universes.
This analysis shows that this exact evolution equation of the curvature
perturbation for the non-flat universes is very complicated, and we need to do
a lot of numerical and analytic work for this new equation in future in order
to judge whether the universe is flat or closed by comparison between theories
and observations.
