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Imagine a two-dimensional version of your lives on some compact, connected surface (orientable or non-orientable). How would you figure out on which surface you are living? Are there experiments you could conduct to determine the orientability and genus of the surface?

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    $\begingroup$ "Time is a flat circle"? $\endgroup$
    – Asaf Karagila
    Apr 7, 2014 at 3:36

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You can look around and record the directions in which you see yourself and at what distance. At least in surfaces of constant curvature, this information should be enough to compute the genus.

This is precisely the idea behind COBE and other later experiments.

Alternatively, you can count for each $\ell\geq0$ the number of simple closed geodesics of length at most $\ell$. As $\ell$ goes to infinity, this grows like $L^{6g-6}$ with $g$ the genus of the surface (provided the genus is at least $2$ and the curvature is constant and equal to $-1$). This is a theorem of Mirzakhani.

Yet another altenative is to pick a point in the surface and construct its cut locus in the sense of Riemannian geometry. It can be shown than the homology of the surface (and therefore the genus) can be computed from that information, and this can be done numerically from experimentral data —this is one of the points of persistent homology; googling for these keywords should bring up lots of information.

Later (14/8/2014) It should be noted that that the beautiful theorem of Maryam Mirzhakhani to which I referred above is part of the reason she got the Fields medal a couple of days ago :-)

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  • $\begingroup$ What's the capital $L$? Just a constant? $\endgroup$ Apr 10, 2014 at 0:53

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