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In string theory there are more than 3+1 dimensions of the space-time, but this is only at very very small "scales", and the concept of dimension used in physics are the one of manifolds because this set the dimension of n-Euclidean space $E^n$ as n. And this is a topological dimension and all the tpological dimensions that i know satisfy the subspace theorem that say that every subspace have dimension less or equal of the space. In some occasions the physicist describe this "phenomena" as the dimensions are "curled-up" and many persons (in the web) like in this question "Curled-up dimensions"? think that the extra "curled-up" dimensions are the dimensions of the fibre a fibre budle but i'm understand this interpretation as that the perceivable space-time of 3+1 dimensions is the base space and the entire space-time is the fibre budle with some fibre. But this interpretation seems to me that not explain the "small scale" extra dimensions, but extra dimensions as in a embedding of the peceivable space time in the entire space-time and that thi is globally with more dimension and not locally. In any case as a analogy sometimes physicists compare a torus and that if you go away from it, it seems to be a circle, and with this i reminded coarse geometry, wich intuitivley is about the global properties of a "space" and that "look the same from afar" is an example.

is perhaps the curled dimension is about coarse geometry? or is topological? and how? or what type of dimesion is and how can a type of "space" have more dimensions in a more small "portion" of the space than in a bigger one?

Thanks in advance.

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I'm not sure I conpletel understand but check out Tensors and Non-Euclidean space - both deal with "curved" space. –  user26649 Jul 18 '12 at 2:48
    
I'm also puzzled by this idea of small scale dimensions. I don't know the answer to this, but I think what might be meant by this is that the space time of physics is a bundle of some sorts and what we see corresponds to the base space of this bundle. I have no idea wether this is in any way what they mean. But I know that principal bundles play some role in the mathematical description of physics. –  Olivier Bégassat Jul 18 '12 at 3:51
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2 Answers

When physicists say that the large-scale space appears 4 dimensional, they are not saying that it is mathematically 4 dimensional. In fact, they are saying that the space is really 10 (or 11, or in some earlier models 26) dimensions, but that the other 6 (or N) dimensions "don't contribute significantly to the large scale features of dimensionality".

To see what this might mean, just look at the surface of a cylinder where the cross-section radius is << 1 in some unit (say it's 10^-40 as an example) and the length/height/whatever is >> 1. Now take a particle on the surface of this cylinder and flick it off in some random direction. Now measure the position of the particle using a "ruler" with an accuracy of 0.1. You couldn't even see the "other", cross-sectional, dimension with that ruler. It couldn't even be turned in that direction in that space. In relation to that ruler, the space would appear one-dimensional and the particle would appear to be moving along a line, even though the space was actual a 2-dimensional surface.

Now consider this with 4 large dimensions and the rest of the dimensions very small. These other dimensions affect the degrees of freedom of the space, which in turn affect the expansions available to the perturbation theory of the action in the quantum field theory and allow for the topological expansions to converge and give interesting theory. But they would not be measurable using tools of poor enough accuracy, and so might have been missed in past observations of the universe.

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and what mean that don't contribute significantly to the large scale features of dimensionality in a mathematical rigorous way? –  Fenrir Jul 18 '12 at 4:10
    
This has nothing to do with fiber bundles or any mathematical description of space attachment beyond simple products on which all dimensions are built. It only means d << D, which has two consequences. 1) measuring devices built at scale D won't see the other dimensions, and 2) there are some places in the theory where perturbative expansion in size makes sense. That's all it means, mathematically. –  ex0du5 Jul 18 '12 at 14:44
    
@EduardoAlanDávalosPeña: Fiber bundles are how fields are associated to a space, if that part of the theory has been confusing. Either that, or they are they are used in the standard ways of mathematics to classify solution spaces and prove local/global correspondence features. –  ex0du5 Jul 18 '12 at 14:50
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Check out Riemann Curvature Tensors and Non-Euclidean space

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