Form of line element of a torus.

In summary, there are different types of tori, including an "ideal torus" and an "embedded torus", each with their own distinct metric. The line element for an "ideal torus" is ds^2=r^2(d\theta^2_1+d\theta^2_2), while the line element for an "embedded torus" can vary depending on its embedding in a higher-dimensional space. The geometry of an "ideal torus" is flat, while the geometry of an "embedded torus" can be curved.
  • #1
arroy_0205
129
0
I noticed somewhere the line element of a two-dimensional torus is written in the form
[tex]
ds^2=r^2(d\theta^2_1+d\theta^2_2)
[/tex]
The author only states that he assumes same radius parameter for simplicity and no further explanation is given. But I do not understand how that form is possible. I find, in such a case the line element should be
[tex]
ds^2=r^2 d\theta^2_1+r^2(1+\cos\theta_1)^2d\theta^2_2)
[/tex]
I cannot reduce this form to the first form. Can anyone explain how the first form for 2D line element of a torus is possible?
 
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  • #2
ideal torus

arroy_0205 said:
I noticed somewhere the line element of a two-dimensional torus is written in the form
[tex]ds^2=r^2(d\theta^2_1+d\theta^2_2)[/tex]
The author only states that he assumes same radius parameter for simplicity and no further explanation is given. But I do not understand how that form is possible. I find, in such a case the line element should be
[tex]ds^2=r^2 d\theta^2_1+r^2(1+\cos\theta_1)^2d\theta^2_2)[/tex]
I cannot reduce this form to the first form. Can anyone explain how the first form for 2D line element of a torus is possible?

Hi arroy_0205! :smile:

Your [tex]ds^2=r^2 d\theta^2_1+r^2(1+\cos\theta_1)^2d\theta^2_2[/tex] is for an ordinary common-or-garden torus, in which the "inside equator" is shorter than the "outside equator".

I think [tex]ds^2=r^2(d\theta^2_1+d\theta^2_2)[/tex] is for an "ideal torus", not embedded in any higher-dimensional space, in which all "equators" have the same length. :smile:
 
  • #3
Hi tiny-tim,
Thanks for your response, but I do not completely get your point. You are trying to say something profound, that for a torus not embedded in a higher dimensional space... etc, but I cannot visualize a torus that way. Can you help? also I have taken the two radii same, that is why there is only one r in the line element that I wrote. My doubt is how embedding and the issue of two radii (same or different) are connected in this case. What is an "Ideal torus" by the way?

Also, do you mean that for a 2-torus which is [tex]S^1\times S^1[/tex], we can simply write
[tex]
ds^2=r^2(d\theta^2_1+d\theta^2_2)
[/tex]
and as generalization, for a n-torus,
[tex]
ds^2=r^2(d\theta^2_1+\cdots+d\theta^2_n)
[/tex]
will this be correct?
 
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  • #4
Is the geometry of torus flat or curved?
If it's flat, why it is flat?
 
  • #5
Hi arroy_0205! :smile:
arroy_0205 said:
Also, do you mean that for a 2-torus which is [tex]S^1\times S^1[/tex], we can simply write …

Yes for a 2-torus, not sure about an n-torus. :smile:
Thanks for your response, but I do not completely get your point. You are trying to say something profound, that for a torus not embedded in a higher dimensional space... etc, but I cannot visualize a torus that way. Can you help? also I have taken the two radii same, that is why there is only one r in the line element that I wrote. My doubt is how embedding and the issue of two radii (same or different) are connected in this case. What is an "Ideal torus" by the way?

An "ideal torus" is S1 x S1.

It's like an Asteroids screen, or a piece of paper with opposite edges identified.

If the screen is square, there's only one r, it it's a rectangle, there are two rs.

The geometry of S1 x S1 is flat (locally identical to a Euclidean plane), so it can't be "joined up" in any higher dimensional Euclidean space.
kahoomann said:
Is the geometry of torus flat or curved?
If it's flat, why it is flat?

Hi kahoomann! :smile:

The geometry of S1 x S1 is flat, because its geometry is locally identical to the Asteroids screen! :biggrin:

(Similarly, the geometry of the surface of a cone is flat.)

The geometry of a common-or-garden torus is curved.
 
  • #6


tiny-tim said:
I think [tex]ds^2=r^2(d\theta^2_1+d\theta^2_2)[/tex] is for an "ideal torus", not embedded in any higher-dimensional space, in which all "equators" have the same length. :smile:

I think this business about ideal versus garden-variety tori is a non-issue.A common definition of an n-dimensional torus is simply a topological space that is diffeomorphic to the space [tex]S^1\times...\times S^1\subset R^{2n}. [/tex] When one says "a torus," usually (depending on the context) this is the 2-dimensional version, [tex]S^1\times S^1\subset C\times C[/tex], using the embedding [tex]S^1\subset C[/tex].

From the definition, one can see that there is no *one* Riemannian metric on a torus (what is being called a line element here), but many. This metric [tex]ds^2=r^2(d\theta^2_1+d\theta^2_2)[/tex] is the flat metric on the torus using as coordinates the angles in the embedding [tex]S^1\times S^1\subset C\times C[/tex]. But the other one mentioned in the original post looks to me like the metric on the torus induced from the Euclidean metric when the torus is embedded in 3-space in the standard way. This is decidedly not a flat metric on the torus.
 
  • #7


Doodle Bob said:
From the definition, one can see that there is no *one* Riemannian metric on a torus (what is being called a line element here), but many. This metric [tex]ds^2=r^2(d\theta^2_1+d\theta^2_2)[/tex] is the flat metric on the torus using as coordinates the angles in the embedding [tex]S^1\times S^1\subset C\times C[/tex]. But the other one mentioned in the original post looks to me like the metric on the torus induced from the Euclidean metric when the torus is embedded in 3-space in the standard way. This is decidedly not a flat metric on the torus.

Hi Doodle Bob! :smile:

But one could equally say that there is no *one* Riemannian metric on a sphere:

we can easily impose a metric on a sphere that "makes it an ellipsoid".

So when we define a sphere, we include the standard metric.

And when we define a torus, we should also include the metric.

An "ideal" torus and an "embedded" (common-or-garden) torus (with the second line element mentioned in the original post) are two different metric spaces, just as a sphere and an ellipsoid are. :smile:
 
  • #8


tiny-tim said:
An "ideal" torus and an "embedded" (common-or-garden) torus (with the second line element mentioned in the original post) are two different metric spaces, just as a sphere and an ellipsoid are. :smile:

But, your so-called "ideal" torus *is* "embedded" too; it's embeddable in R^4 as [tex]S^1\times S^1[/tex] and the metric that is induced by this embedding from the 4-dimensional Euclidean metric is, in fact, the standard flat metric on the torus, i.e., the Asteroids metric.

"Torus" is really reserved as topological term rather than a geometrical one. "Flat torus," e.g., is a better way of conveying a specific geometry rather than "ideal" since "ideal" conveys either a Platonistic point of view, which doesn't really to apply in this matter, or an algebraic point of view, which is equally as irrelevant here.
 
  • #9


Doodle Bob said:
But, your so-called "ideal" torus *is* "embedded" too; it's embeddable in R^4 as [tex]S^1\times S^1[/tex] and the metric that is induced by this embedding from the 4-dimensional Euclidean metric is, in fact, the standard flat metric on the torus, i.e., the Asteroids metric.

Hi Doodle Bob! :smile:

So even though all its sides are curved, the geometry is flat, in exactly the same way as the surface of a cylinder in R3 is flat?

:confused: mmm … I never could think in four dimensions. :confused:

Yes, I suppose it does work in R4, because R4 has the direct product geometry R2 x R2, which S1 x S1 naturally slips into,

while S1 x S1 doesn't fit in with the direct product R3 = R2 x R1.

(I see you've had to put people right on this https://www.physicsforums.com/archive/index.php/t-80846.html".)

So if I say "embedded" in future … I'd better specify "in R3".

Thanks for the correction! :smile:
 
Last edited by a moderator:

1. What is a torus?

A torus is a geometric shape that resembles a donut or a tire. It is a three-dimensional object with a circular cross-section and a hole in the center.

2. What is a line element?

A line element is a mathematical concept that represents a small segment of a curve or surface. It is used to calculate properties such as length, area, and volume.

3. How is the form of line element of a torus expressed?

The form of line element of a torus is expressed using the parametric equations for a torus. These equations define the position of any point on the surface of the torus in terms of two parameters, typically denoted by u and v.

4. What does the line element of a torus look like?

The line element of a torus is a small segment of the torus's surface, which can be approximated by a curved rectangle. It has a length, width, and height, and its curvature depends on the shape and size of the torus.

5. How is the line element of a torus used in calculations?

The line element of a torus is used in calculations to find properties such as the surface area and volume of a torus. It is also used in the study of differential geometry and in the development of mathematical models for physical systems.

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