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In standard 3+1 dimensional spacetime, the metric tensor is of order 4 and had ten independent coefficients, hence there are 6 terms off the diagonal in the corresponding $4\times 4$ real symmetric matrix. On the other hand, superstring theory postulates that there are 6 additional spatial dimensions that have to be compactified in a Calabi-yau manifold. My question is thus: are these two facts somehow related or is it just mere coincidence? For example, would the observable dimensions correspond to eigenvalues of the metric tensor?

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    $\begingroup$ I strongly suspect this is a coincidence. Questions like this run the risk of being classified as "numerology". $\endgroup$ – user_of_math Jul 23 '14 at 15:39
  • $\begingroup$ But then M theory postulates there are 11 dimensions, so that's 7 compact ones but only 6 off diagonal terms. $\endgroup$ – John Rennie Jul 23 '14 at 16:14
  • $\begingroup$ @John Rennie: if the metric tensor depends on a continuous parameter, like, say, the resolution, then 11 numbers are needed to characterize it. I guess further development of scale relativity theory will that indeed the metric tensors depends on the resolution. $\endgroup$ – Sylvain JULIEN Jul 24 '14 at 10:58

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