  In a Universe with a detectable nontrivial spatial topology the last
scattering surface contains pairs of matching circles with the same
distribution of temperature fluctuations --- the so-called circles-in-the-sky.
Searches undertaken for nearly antipodal pairs of such circles in cosmic
microwave background maps have so far been unsuccessful. Previously we had
shown that the negative outcome of such searches, if confirmed, should in
principle be sufficient to exclude a detectable non-trivial spatial topology
for most observers in very nearly flat ($0<\mid\Omega_{\text{tot}}-1\mid
\lesssim10^{-5}$) (curved) universes. More recently, however, we have shown
that this picture is fundamentally changed if the universe turns out to be {\it
exactly} flat. In this case there are many potential pairs of circles with
large deviations from antipodicity that have not yet been probed by existing
searches. Here we study under what conditions the detection of a single pair of
circles-in-the-sky can be used to uniquely specify the topology and the
geometry of the spatial section of the Universe. We show that from the
detection of a \emph{single} pair of matching circles one can infer whether the
spatial geometry is flat or not, and if so we show how to determine the
topology (apart from one case) of the Universe using this information. An
important additional outcome of our results is that the dimensionality of the
circles-in-the-sky parameter space that needs to be spanned in searches for
matching pair of circles is reduced from six to five degrees of freedom, with a
significant reduction in the necessary computational time.
