Exploring Torsion in 2 Dimensions: A Differential Geometry Perspective

[bcrowell]

This is really more of a differential geometry question than a GR question. Is it possible to have torsion in two dimensions? I've seen statements implying that torsion is only of interest in 3 or more dimensions, but I don't see why. My understanding is that in two dimensions, you could not have a totally antisymmetric torsion tensor, and therefore it's not possible to have torsion in which parallel-transporting a tangent vector along a geodesic keeps it tangent to the geodesic. This doesn't seem the same as saying that torsion is impossible or of no interest in 2 dimensions.

 

[haushofer]

Maybe i m overlooking something, but torsion is just the antisymm. part of the connection. So why can't that exist in 2 dim.?

 

[TrickyDicky]

This is really more of a differential geometry question than a GR question. Is it possible to have torsion in two dimensions? I've seen statements implying that torsion is only of interest in 3 or more dimensions, but I don't see why.

This is an ambiguous question as worded. What do you mean by torsion in 2D? Certainly it is possible to talk about torsion of 2D surfaces in 3 or n dimensions. A different thing is the possibility of torsion of curves inside a 2D space, there is no such thing.

My understanding is that in two dimensions, you could not have a totally antisymmetric torsion tensor, and therefore it's not possible to have torsion in which parallel-transporting a tangent vector along a geodesic keeps it tangent to the geodesic.

That is not dependent on the number of dimensions but on the Riemannian connection you are describing. The torsion tensor is a vector-valued two form, when completely anty-symmetrized it vanishes leaving the connection totally symmetric.

 

[bcrowell]

Maybe i m overlooking something, but torsion is just the antisymm. part of the connection. So why can't that exist in 2 dim.?

Yeah, I don't see it either, except that in 2 dimensions you can't have torsion and also have parallel transport of a tangent vector along a geodesic maintain its tangency.

 

[TrickyDicky]

Didn't you see #3?

 

[Ben Niehoff]

Yeah, I don't see it either, except that in 2 dimensions you can't have torsion and also have parallel transport of a tangent vector along a geodesic maintain its tangency.

We had a discussion about "geodesics" and torsion earlier. Which definition of "geodesic" do you mean here? A curve whose tangent is parallel-transported along itself, or a curve that locally extremizes the length functional?

It is clear that the first type of curve exists even in the presence of torsion, although it might no longer agree with the second.

 

[TrickyDicky]

We had a discussion about "geodesics" and torsion earlier. Which definition of "geodesic" do you mean here? A curve whose tangent is parallel-transported along itself, or a curve that locally extremizes the length functional?

It is clear that the first type of curve exists even in the presence of torsion, although it might no longer agree with the second.

I took it to mean the latter type of geodesic (the one determined by Levi-civita connection). But as I said that would be independent of the dimensionality of the manifold. If it is the other type, or any other curve my understanding is that there is no torsion in a two manifold, right?

 

[Ben Niehoff]

I see no reason there should be no torsion in 2 dimensions. However, clearly the totally antisymmetric part of the torsion vanishes.

This means that, assuming metric-compatibility, any given set of geodesics uniquely determines the connection (because the totally antisymmetric part of torsion is what allows many different metric-compatible connections to have the same geodesics).

But this does not mean that the Levi-Civita connection is the only metric-compatible connection. It does mean that any other metric-compatible connection will have different geodesics, as Ben points out.

 

[TrickyDicky]

I see no reason there should be no torsion in 2 dimensions.

I don't understand this statement. So you are sayin that a curve in a surface can have torsion, can you point me to a reference that shows how to calculate it?

 

[bcrowell]

We had a discussion about "geodesics" and torsion earlier. Which definition of "geodesic" do you mean here? A curve whose tangent is parallel-transported along itself, or a curve that locally extremizes the length functional?

The point of the stuff you quoted was basically just that I was trying to make the distinction between those two definitions.

This means that, assuming metric-compatibility, any given set of geodesics uniquely determines the connection (because the totally antisymmetric part of torsion is what allows many different metric-compatible connections to have the same geodesics).

This sounds interesting, but I don't really understand it. Are you a condensing a long, complicated argument that I would need to read in a book to understand it, or is it a simple argument that could be explained here?

 

[TrickyDicky]

I see no reason there should be no torsion in 2 dimensions.

Isn't a planar curve with non-vanishing curvature defined to have zero torsion at each point?
From the Wikipedia entry for torsion:"A plane curve with non-vanishing curvature has zero torsion at all points."
If you are referring to the torsion of surfaces embedded in higher spaces obviously there is torsion for the surface geodesics. That is why I asked the OP to clarify what he meant by "torsion in 2 dimensions".

 

[Ilmrak]

I'm sorry for the intrusion, I'm not understanding how there cannot be torsion in 2 dimensions.

Given a connection $\nabla$ on a differentiable manifold the torsion tensor $T$ is defined by

$T(X,Y) \equiv \nabla_X Y - \nabla_Y X - [X,Y]$,

where $X, Y$ are tangent vector fields. It's antisymmetric by construction and in components it is

$T(X,Y)^\mu \partial_\mu = (\Gamma^\mu_{\alpha \beta} - \Gamma^\mu_{\beta \alpha})X^\alpha Y^\beta \partial_\mu$.

So why should torsion vanish in two dimensions?

Ilm

 

[bcrowell]

I'm sorry for the intrusion, I'm not understanding how there cannot be torsion in 2 dimensions.

Given a connection $\nabla$ on a differentiable manifold the torsion tensor $T$ is defined by

$T(X,Y) \equiv \nabla_X Y - \nabla_Y X - [X,Y]$,

where $X, Y$ are tangent vector fields. It's antisymmetric by construction and in components it is

$T(X,Y)^\mu \partial_\mu = (\Gamma^\mu_{\alpha \beta} - \Gamma^\mu_{\beta \alpha})X^\alpha Y^\beta \partial_\mu$.

So why should torsion vanish in two dimensions?

I'm more comfortable with straight index gymnastics notation, whereas it looks like you're expressing this in a combination of index-gymnastics and index-free notation (?), so I'm not sure how to interpret it.

In the notation I'm more comfortable with, we'd have a three-index tensor $T^c_{ab}$. It's antisymmetric by definition on the indices ab. In $\ge 3$ dimensions, it can also be antisymmetric on all three indices, in which case the connection preserves tangent vectors along geodesics. In two dimensions, it's only possible for a nonzero three-index tensor to be antisymmetric on two of its indices, in which case there are two degrees of freedom, $T^1_{12}$ and $T^2_{12}$. This does *not* imply directly that torsion is impossible in two dimensions.

 

[Ilmrak]

I'm more comfortable with straight index gymnastics notation, whereas it looks like you're expressing this in a combination of index-gymnastics and index-free notation (?), so I'm not sure how to interpret it.

Yes, sorry, I should have written in a more standard notation $T(X,Y) = T^\mu_{\alpha \beta}X^\alpha Y^\beta \partial_\mu$.

In the notation I'm more comfortable with, we'd have a three-index tensor $T^c_{ab}$. It's antisymmetric by definition on the indices ab. In $\ge 3$ dimensions, it can also be antisymmetric on all three indices, in which case the connection preserves tangent vectors along geodesics. In two dimensions, it's only possible for a nonzero three-index tensor to be antisymmetric on two of its indices, in which case there are two degrees of freedom, $T^1_{12}$ and $T^2_{12}$. This does *not* imply directly that torsion is impossible in two dimensions.

So is the point that, in two dimension, there cannot be a connection with non vanishing torsion such that the two definitions of geodesics are equivalent?

Ilm

 

[TrickyDicky]

So is the point that, in two dimension, there cannot be a connection with non vanishing torsion such that the two definitions of geodesics are equivalent?

Not exactly, what happens is that in surfaces the torsion tensor is completely determined by the torsion form, that usually is a 2-form but in surfaces is a one-form.
So this is the sense in which torsion is trivial in 2D, the amount of information it gives is quite limited, basically it gives the orientation of the surface. See relative (geodesic) torsion and the Darboux frame.

 

[Ilmrak]

Not exactly, what happens is that in surfaces the torsion tensor is completely determined by the torsion form, that usually is a 2-form but in surfaces is a one-form.
So this is the sense in which torsion is trivial in 2D, the amount of information it gives is quite limited, basically it gives the orientation of the surface. See relative (geodesic) torsion and the Darboux frame.

I think I will have to study a bit more differential geometry, I don't know anything about the torsion form
Thank you for the answer!

Ilm

 

[TrickyDicky]

I think I will have to study a bit more differential geometry, I don't know anything about the torsion form
Thank you for the answer!

Ilm

You're welcome!
For me the bottom line here is that for surfaces torsion doesn't seem to be intrinsic (it is a covector for n=2 and a full tensor for n>2) in the same way curvature is intrinsic (a scalar for n=2 and a full tensor for n>2), and I guess that is why it might seem less interesting or important.

 

[Ben Niehoff]

Tricky, I think you're talking about something else. You seem to be referring to the torsion of curves, which is an extrinsic curvature that measures the rate at which the curve leaves the osculating plane.

What Ben asked about in the OP is the torsion of a connection, which is an intrinsic curvature that measures twisting of frames under parallel transport.

These are both called "torsion", but they are completely different things.

 

[TrickyDicky]

Tricky, I think you're talking about something else. You seem to be referring to the torsion of curves, which is an extrinsic curvature that measures the rate at which the curve leaves the osculating plane.

What Ben asked about in the OP is the torsion of a connection, which is an intrinsic curvature that measures twisting of frames under parallel transport.

These are both called "torsion", but they are completely different things.

As I said in surfaces the connection skew-symmetric part that constitutes the torsion tensor is determined by only two independent components, the one-form(torsion form) in 2 dimensions. And usually vectors and covectors are not considered invariants in the same way scalars and tensors with more than one index are.

Is there something specifically that you disagree with in the previous paragraph?

 

[Spinnor]

Looking at section 5.8.2 "The torsion tensor", of,

http://www.lightandmatter.com/html_books/genrel/ch05/ch05.html#Section5.8

it looks like the answer is kind of. From the above,

"Torsion that does not preserve tangent vectors will have nonvanishing elements such as τxxy, meaning that parallel-transporting a vector along the x-axis can change its x component. Torsion that preserves tangent vectors will have vanishing τλμν unless λ, μ, and ν are all distinct."

If you include time as a dimension does that allow for torsion with 2 space dimensions?

Edit, I just noticed I referenced bcrowell's work, %^). Its an interesting read, thank you for it!