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Trying to make sense of small and large extra dimension(s) of phyiscal space in a simple intuitive example.

Consider a two dimensional manifold like $\mathbb{R}^2$ and we are trying to add a small and a large extra dimension.

Do we mean by small extra dimension in this case something like $(0,1) \times \mathbb{R}$ (the flat case) or $S^1 \times \mathbb{R}$ (the curved case)?

Do we mean by large extra dimension something like $\mathbb{R^2} \times \mathbb{R}=\mathbb{R}^3$?

Do we mean in the case of our three dimensional space that basically we have a base space of our phyiscal three dimensional space with a total space built by adding a fiber and thus creating a fiber bundle or a even more general an arbitrary total space?

Does the extra dimension need to be real or can we even consider the complex manifolds, in the case of adding extra dimension to the phyiscal space, for example $\mathbb{C} \times \mathbb{R^3}$ or (Riemann surface) $\times \mathbb{R^3}$

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  1. The word large and small are determined by a metric tensor/characteristic length scale. Small dimensions can typically not be detected by current experiments. Large dimensions are typically of cosmological size, however, see e.g. the ADD model.

  2. The topology of extra dimensions need not be compact: E.g. the metric tensor could in principle contain a warp factor that makes the extra dimension effectively small.

  3. The full spacetime is not necessarily a total space of a fiber bundle.

  4. Extra dimensions may have a complex structure, such as e.g. a Calabi-Yau manifold.

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