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I’d like to add some sentences describing a geometric understanding of the “torsion” of a connection on a Riemannian manifold to torsion. But perhaps my understanding of torsion is wrong, so I’m running it by you guys first. I wrote it down on math overflow and I’m curious what people think.
There’s probably quite a lot of material worth borrowing from that thread. What about organising things as Chris Schommer-Pries does at a more general level than Riemannian geometry?
By the way, what Chris S.-P. indicates in that entry on the integrability of G-structures is alluding to what is explained in section 2.6 of
This perspective reminds me of what I was recently after on MO here,
But as far as I can see, for Bruce’s purposes the perspective of integrability of G-structures is going a bit off on a tangent. I think specifically for torsion of metric connections in the context of gravity, there is a clear best perspective, and its the one indicated on the nLab at torsion of a metric connection:
namely a metric connection is precisely a Cartan connection for the inclusion of the orthogobal/Lorentz group into the Euclidean/Poincaré group. Such a connection has precisely two different curvature components, corresponding to the two pieces in the extension
$O(d) \longrightarrow Iso(\mathbb{R}^d) \longrightarrow \mathbb{R}^d \,.$The two curvature components are the Riemann curvature and the torsion.
And this is pretty much what Bruce is after, I think.
David: I agree there’s lots of stuff going on there. In the long term I’m going to try understand CSP’s comment. Right now I’m after a more down-to-earth geometric interpretation. Urs: That’s interesting I will have to think about that.
I think there are some problems with my interpretation of torsion. For instance, it would imply that a metric connection on a surface cannot have torsion (because once you fix a vector in a frame $e_1, e_2$, you cannot rotate the remaining vector). I may have been making the error that it is only in odd dimensions that a rotation has an “axis of rotation”.
I’m going to try understand CSP’s comment
It’s Dominic Joyces’s observation, explained in section 2.6 of his book “Compact manifolds with special holonomy”
Urs: That’s interesting I will have to think about that.
You must have thought of that already, as this is Cartan’s definition of torsion.
I may have been making the error that it is only in odd dimensions that a rotation has an “axis of rotation”.
No, an axis of rotation only exists precisely in 3 dimensions. In all other dimensions the axis is replaced by the corresponding Poincaré dual of the given surface of rotation.
Yes, apologies for getting rotations wrong. Regarding Cartan’s definition of torsion - I have to admit that I haven’t thought much about the generalized case of Cartan connections, I mainly only know about connections on a principal bundle which is a special case I know. I don’t berate myself for this, since Cartan connections aren’t dealt with in the standard books on differential geometry (eg. Spivak). But you’re right, I’m sure they give a nice explanation of torsion. I will just need some time to get my head around that.
A Cartan connection is just a principal connection subject to the constraint that at each point it identifies the tangent space with the quotient Lie algebra.
Textbooks may not have it, but the nLab does: Cartan connection. ;-)
It’s super-simple. But it’s also true that there are many discussions of Cartan connections out there which make it look more intricate.
In particular, physicists have been using Cartan connections for ages, without using the term. This is just what the first order formulation of gravity is: an $Iso(d-1,1)$-connection such that it identifies at each point the tangent space of spacetime with an $\mathbb{R}^d$. That’s all.
And torsion is just the part of the curvature 2-form with values in the $\mathbb{R}^d$-factor. This statement is manifestly Cartan’s (and hence the standard) definition of torsion.
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