I What's the name of a 2-torus looped in the 4th dimension?

[greswd]

To form a 2-torus, a narrow tube can be bent into a loop and joined end to end:

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But instead of forming this loop in our three-dimensional space, the loop can also be formed in a direction perpendicular to three-dimensional space, moving it into the fourth dimension of space.

What's the name of this object?
Its similar to how the Klein bottle is a four-dimensional object, and this thing that we see is a poor three-dimensional representation

 

[andrewkirk]

What you have written is not a topological specification, and cannot be interpreted unambiguously. For example, it is possible for a standard torus to be embedded in 4D space in such a way that only parts of it are in 'our' 3D space, and the topology of that torus is no different from one that is embedded in 'our' space.

The standard way to specify a simple 2D surface without boundary is by drawing a square and then showing by arrows on the four edges of the square, how the edges are pasted together to form the surface. There are three possibilities: torus, klein bottle and projective plane. The latter two cannot be embedded in 3D space, and are non-orientable.

This page has a good explanation of the edge-pasting approach to specifying simple 2D surfaces.

 

[greswd]

For example, it is possible for a standard torus to be embedded in 4D space in such a way that only parts of it are in 'our' 3D space, and the topology of that torus is no different from one that is embedded in 'our' space.

sorry, can you elaborate on this? thanks

 

[andrewkirk]

Just take a complete, standard torus and rotate it in a direction that is perpendicular to our 3D space. It will still be a standard torus, but only slivers of it will remain in our space. But the key point is that the description in the OP does not specify a topological figure because it does not supply the necessary information.

 

[greswd]

Just take a complete, standard torus and rotate it in a direction that is perpendicular to our 3D space. It will still be a standard torus, but only slivers of it will remain in our space.

slivers meaning just infinitely thin flat planes remaining in 3D space?

But the key point is that the description in the OP does not specify a topological figure because it does not supply the necessary information.

If a straight tube is bent into a torus, the inner (red) region will be compressed, while the outer (blue) region will be stretched.

But if the loop is made in the 4th dimension, neither region will be compressed nor stressed.

That's the concept I have in mind.

Similar to the example of the Klein bottle I mentioned, try to form it in three dimensions and it will self-intersect, only in four dimensions can be it free of that.

And so for the torus, it can be free of compressing and stretching by going "up" into the fourth dimension.

 

[Infrared]

If a straight tube is bent into a torus, the inner (red) region will be compressed, while the outer (blue) region will be stretched.

But if the loop is made in the 4th dimension, neither region will be compressed nor stressed.

That's the concept I have in mind.

It sounds like you're describing a flat torus?

 

[greswd]

It sounds like you're describing a flat torus?

Seems like it. Is it the simplest way to form a torus free from compressing and stretching in the 4th dimension?

 

[lavinia]

Seems like it. Is it the simplest way to form a torus free from compressing and stretching in the 4th dimension?

One can make a flat torus in 3 dimensions but not one that is smoothly embedded. The unit normal will be continuous but not differentiable. Here is a picture. While it may not be obvious, there is no stretching or compressing here.


A theorem of Hilbert says that any smoothly embedded closed surface in 3 space must have a point of positive curvature. At such a point the surface is stretched.

A Klein bottle can also be made without stretching or compressing. Here is an immersion of a flat Klein bottle in 4 space where the self-intersection in your picture still occurs.

https://www.ams.org/journals/bull/1941-47-06/S0002-9904-1941-07501-4/S0002-9904-1941-07501-4.pdf

In high enough dimensions, not sure how high, one can make a flat Klein bottle without self-intersection.
If you allow stretching and compressing , then a Klein bottle without self intersection can be made in four dimensions.

Thinking about post #2, @andrewkirk describes how to make a torus and a Klein bottle without any need for 3 or 4 space or for that matter any ambient space at all. It gives an intrinsic description of how these surfaces are put together. It is a separate question whether surfaces - or more generally manifolds of higher dimension - can be incarnated in Euclidean space of some dimension.

 

[zinq]

Yes, that flat torus in 3-space was described in this paper https://www.pnas.org/content/109/19/7218 in 2012. It was, I think, the first example of an explicit C¹ embedding of a flat compact surface in R³, as shown possible by John Nash and Nicolaas Kuiper.

But as for the original question: Possibly the simplest torus in 4-space is just the (also flat) cartesian product of two circles, such as {(cos(s), sin(s), cos(t), sin(t)) ∈ R⁴ | 0 ≤ s, t ≤ 2π}. It's easy to see that this lies in the unit 3-sphere S³ in R⁴ of radius √2, and this is clearly an infinitely differentiable embedding, even real analytic.