  We study the effects of the spatial curvature on the classical and quantum
string dynamics. We find the general solution of the circular string motion in
static Robertson-Walker spacetimes with closed or open sections. This is given
closely and completely in terms of elliptic functions. The physical properties,
string length, energy and pressure are computed and analyzed. We find the {\it
back-reaction} effect of these strings on the spacetime: the self-consistent
solution to the Einstein equations is a spatially closed $(K>0)$ spacetime with
a selected value of the curvature index $K$ (the scale f* is normalized to
unity). No self-consistent solutions with $K\leq 0$ exist. We semi-classically
quantize the circular strings and find the mass $m$ in each case. For $K>0,$
the very massive strings, oscillating on the full hypersphere, have $m^2\sim K
n^2\;\;(n\in N_0)$ {\it independent} of $\alpha'$ and the level spacing {\it
grows} with $n,$ while the strings oscillating on one hemisphere (without
crossing the equator) have $m^2\alpha'\sim n$ and a {\it finite} number of
states $N\sim 1/(K\alpha').$ For $K<0,$ there are infinitely many string states
with masses $m\log m\sim n,$ that is, the level spacing grows {\it slower} than
$n.$ The stationary string solutions as well as the generic string fluctuations
around the center of mass are also found and analyzed in closed form.
