Topology Q: Curved Space w/out Higher Dimensions?

[Pavel]

Hi, does it make sense to posit a curved (hall of mirrors type) space without higher dimensions? In other words, if I say we live in a 3 dimensional torus shaped universe, does that statement necessarily entail there's at least a 4 dimensional overall hyperspace? Or can I have a torus curved into itself by itself, without it being in a higher dimensional context?

Thanks!

 

[CompuChip]

There are two kinds of curvature: intrinsic and extrinsic. Intrinsic curvature depends only on the space and not on the embedding of that space into some higher-dimensional space. In fact, though it is a theorem that every <some weak properties here> topological space can be embedded in a Euclidean "flat" space, we usually try to define all the relevant properties without reference to such an ambient space, so we can really view it as an object of its own (though, frankly, I have trouble imagining a two-dimensional surface without using three dimensions).

 

[jonmtkisco]

Hi CompuChip,

Intrinsic curvature depends only on the space and not on the embedding of that space into some higher-dimensional space. In fact, though it is a theorem that every <some weak properties here> topological space can be embedded in a Euclidean "flat" space, ...

When one says that, for example, an "intrinsically curved" 3-dimensional surface can be embedded in a Euclidean space, does "embedding" refer solely to a mathematical property, or is it also a tangible physical property?

How can it be a tangible physical property if we can't actually construct a solid 3-dimensional physical model that has simple 3-dimensional surfaces?

It seems to me that any physical embodiment of a 3-dimensional surface unquestionably requires the physical existence of a 4th spatial dimension.

Jon

 

[CompuChip]

"Embedding" is indeed well-defined mathematically, but that definition is such that it corresponds to our intuition. For example, a two-dimensional surface can often be embedded in three-dimensional Euclidean space. In that case, we can actually make a physical model of it which we can touch, etc. Of course, since we only have three dimensions available, it is not possible to physically construct a manifold which can only be embedded in spaces of dimension > 3. So the mathematical possibility of embedding in higher dimensional spaces says nothing about the physical possibility, for example: one can construct smooth manifolds of arbitrary high dimensions (just think about the n-sphere, which can be embedded in an (n + 1)-dimensional Euclidean space) without ever being able to really visualize them. The fact that we can construct a 120435 dimensional manifold of course does not at all imply the existence of 120436 dimensions.

Off-topic, that last remark reminds me of this joke

A mathematician and his best friend, an engineer, attend a public lecture on geometry in thirteen-dimensional space. "How did you like it?" the mathematician wants to know after the talk. "My head's spinning," the engineer confesses. "How can you develop any intuition for thirteen-dimensional space?" "Well, it's not even difficult. All I do is visualize the situation in n-dimensional space and then set n = 13."
Taken from the first source found by Google[/size]

 

[jonmtkisco]

CompuChip, I think the joke is very much on-topic! Thanks.

Jon

 

[Pavel]

CompuChip, thank you very much for the explanation. Loved the joke too :)

Pavel