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The Physical Constant that appears every where might be possible that related to geometry of the universe that still needed to be uncovered. For example, if a person is confined inside a room which in turn is like a closed system, the events taking place within the walls of the room must be related to dimensions of the room, its geometry and size. Similarly, the fundamental constants that we encounter in nature, in our text book might come out of some geometry which the universe has but which is not known and since these constants are constants the geometry that universe must also be fixed.

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Of course, universe has a geometry, but all the details of these geometry are not known yet. The best description of fundamental physics is given by String Theory (or M-theory), which states that microscopically our spacetime has the form of $M \times CY$, where $M$ is a microscopically observable spacetime having the form of de Sitter space or close to it (see this Wiki article on de Sitter cosmology), and $CY$ is a particular 6-dimensional compact manifold, called the Calabi-Yau manifold.

According to String Theory, there is just one dimensionful fundamental constant, namely, tension of fundamental string, while all the other coupling constants and masses of particles (as well as their number, etc.) are expressible in terms of string tension, expectation values of scalar fields, and the geometry of compact $CY$. Currently, the precise form of $CY$ is not known, as well as the dynamical principle which would allow us to find it from the first principles. See the classical (though obsolete in some respects) textbook "Superstring Theory" by Green, Schwarz and Witten, in which compactifiactions of String Theory are described in great details.

String Theory predicts many other interesting modifications of pre-stringy geometry. For example, it is known that if certain field predicted by String Theory (so-called $B$-field) is non-zero, spacetime geometry becomes non-commutative (see the original paper of Seiberg and Witten on the subject). Another interesting modification of geometry is related to T-duality (or Mirror Symmetry), an exact equivalence of different manifolds, on which String Theory is compactified.

The subjecct is immense, so if you are interested, I recommend you to read a book on it. For example, "D-branes" by Johnson or "Introduction to String Theory and M-theory" by Becker, Becker and Schwarz.

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    $\begingroup$ Currently, it is not even known whether the compactified part is a Calabi-Yau (and for M-theory the compact part would be seven-dimensional, so not Calabi-Yau but G2). We do not observe supersymmetry so far so there's no absolutely convincing reason for the compactification to preserve supersymmetry (which is what being a fluxless Calabi-Yau corresponds to), and there are fluxed compactifications that might preserve supersymmetry, too. $\endgroup$ – ACuriousMind Jan 30 '17 at 15:14
  • $\begingroup$ @ACuriousMind Yes, of course. I gave just a very basic presentation of the subject. $\endgroup$ – Andrey Feldman Jan 30 '17 at 15:39
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The universe has some geometry: (1) topologically, it is almost flat (in the three-dimensional sense, not flat like a disk), (2) it is finite, (3) it is expanding (4) at an accelerating rate, (5) it has some properties that are slightly scale dependent and (6) it has a cell-like matter distribution (7) that is more or less homogeneous at a large enough scale.

The constant that is most obviously related to the geometry of the universe is Hubble's constant, which is only fundamental in a loose sense of the term.

The cosmological constant could also be related to the geometry of the universe.

There are some proposed modifications to general relativity in which Newton's constant, G, is renormalizable and "runs" with energy scale which in some sense corresponds to the geometry of the universe.

There is no good reason to think that any of the constants of the Standard Model are related to the geometry of the universe, and precision tests of changes in the electro-magnetic coupling constant in the early days of the universe based upon astronomy observations show that it has not changed over time as it might if it were dependent upon the geometry of the universe.

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    $\begingroup$ I think that $\pi$ also could be highly rated for being the most obviously related to the geometry of the universe. $\endgroup$ – user130529 Jan 30 '17 at 10:24
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    $\begingroup$ If by almost flat you mean almost Ricci flat, then this curvature is not a topological invariant, so one would describe it as a property of its geometry rather than 'topologically.' $\endgroup$ – JamalS Jan 30 '17 at 10:40

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