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

What can the detection of a single pair of circles-in-the-sky tell us about the geometry and topology of the Universe ?