Hi all, I am working through Visser's notes http://msor.victoria.ac.nz/twiki/pub/Courses/MATH465_2012T1/WebHome/notes-464-2011.pdf section 3.5 onward. I am trying to differentiate between the torsion and the Riemann curvature tensor in a heuristic manner. It appears from "Geometric interpretation 1" (page 73) that the torsion tensor is related to going along both paths of a parallelogram to get to opposite corners, and then finding the difference...how the infinitesimal parallelogram does not close. The Riemann tensor is introduced simply through the noncommutativity of the the covariant derivative, though looking around on the internet, I have also found references to infinitesimal parallelograms. Could someone explain, in the same sense as above, what motivates the calculation of the Riemann tensor (i.e. commutator of covariant derivatives)? I have heard references to the "twist" of frames along a geodesic for the torsion as opposed to the "roll" for the Riemann curvature --- if you could explain these, that would also be appreciated. Cheers
Hmmm, is it related to the fact that the torsion is to do with the "position not commuting" in the sense that for a parallelogram ABCD, the position reached by going from A-> B -> C is different to that reached by going from A -> D -> C ? Whereas for the Riemann curvature tensor, we parallel transport a vector round an infinitesimal loop, but we return to the same position (with 0 torsion), and it is the change in direction of the vector we are concerned about? Cheers
OK, more looking on my part at the wikipedia article in the first post: http://en.wikipedia.org/wiki/Riemann_curvature_tensor#Geometrical_meaning. Does anyone know how to show the equation [tex]\left.\frac{d}{ds}\frac{d}{dt}\tau_{sX}^{-1}\tau_{tY}^{-1}\tau_{sX}\tau_{tY}Z\right|_{s=t=0} = (\nabla_X\nabla_Y - \nabla_Y\nabla_X)Z \ ?[/tex] Cheers
Thanks Matterwave, I remember reading the thread a while ago, but seeing it in a new light helped. In saying that, while I agree with your construction in post#4, I always worry that such "head to tail" diagrams imply a distance, when in this case, I believe it is more about the direction. In your diagram, what I think would be the case with torsion is that [itex]\nabla_X Y - \nabla_Y X[/itex] is the direction composed of that of [itex][X,Y][/itex] as in your post, plus the direction due to the parallelogram not quite closing in terms of positions, as in Visser's notes, which is due to torison. (I.e. torsion is related to differences in position, not direction.) That's not clear, but look at Visser pg 73-74, and you'll see what I mean. To quote him "in a manifold with torsion there are no infinitesimal parallelograms". Ben Niehoff's answer about torsion being a twist was helpful, though I'd need to think about it more to write the equation he talked of. As for the "rolling" associated with the Riemann tensor, I'm happy to hear thoughts. As for my question, I am now more interested in how to prove [tex]\left.\frac{d}{ds}\frac{d}{dt}\tau_{sX}^{-1}\tau_{tY}^{-1}\tau_{sX}\tau_{tY}Z\right|_{s=t=0} = (\nabla_X\nabla_Y - \nabla_Y\nabla_X)Z [/tex] as I think I can do the interpretation once I've covered this. Any thoughts? I don't see why it should the product of propagators becomes a product and subtraction of covariant derivatives, i.e. I'd have thought it'd be [itex]\nabla_X+\nabla_Y - \nabla_Y-\nabla_X[/itex] or [itex]\nabla_X\nabla_Y (-\nabla_Y)(-\nabla_X)=\nabla_X\nabla_Y\nabla_Y\nabla_X[/itex], instead of a mixture of both. This reminds me of how the commutator of a lie algebra comes from the commutator of the lie group. Still not sure. Ideas? Cheers
Ahah, Figure 1 in this may help explain what I was talking about: http://arxiv.org/PS_cache/arxiv/pdf/0711/0711.1535v1.pdf
I am confused as to what [itex]\tau_{sX}^{-1}\tau_{tY}^{-1}\tau_{sX}\tau_{tY}Z[/itex] even means. Assuming Z belongs to TpM, why would [itex]\tau_{sX}^{-1}\tau_{tY}^{-1}\tau_{sX}\tau_{tY}Z[/itex] belong to TpM (so that differentiation makes sense)?
That's the same figure as in my thread except torsion is present and so the "inner parallelogram" doesn't close, you would agree? I am not familiar with the equation you put up, so I probably can't help you there. I've seen many other ways to describe the curvature, the most geometric of which is parallel transport around a closed loop. However, I am used to using covariant derivatives rather than propagators.
If you look at the wiki link I gave, this is just parallel propagation of a vector [itex]Z[/itex] around a parallelogram with sides [itex]X,Y[/itex].
Sure, I'll agree with you on that count. Covariant derivatives and parallel propagators go hand in hand. As in the post above, the equation I am referring to is parallel propagation around a "parallelogram" (I am not sure if it closes with torsion etc). As for the argument as why [itex][\nabla_X , \nabla_Y]Z[/itex] is important for curvature, I only half understand it. You are looking at how the vector [itex]Z[/itex] changes in the [itex]X[/itex] direction, and seeing how this changes in the [itex]Y[/itex] direction, and then subtracting the reverse. It doesn't exactly seem intuitive as to why this corresponds in some way to curvature --- at least to me. Two points as an aside: 1) Is it propagation "around" a parallelogram (i.e. back to the start) or taking the two different routes from diagonal corners, and subtracting (as [itex][\nabla_X , \nabla_Y]Z[/itex] and your diagram imply)? This is part of my problem with the formula I am trying to work out: the LHS appears to be around the parallelogram, the RHS subtracting diagonal routes. 2) It seems like torsion is first order in covariant derivatives, and the riemann tensor second order. Do we have a third order? I.e. using a parallelepiped with three vectors? Cheers
The idea is that in the presence of curvature, a vector which is parallel transported around a closed loop does not return to itself. We use that as a DEFINITION of curvature. In a flat space, parallel transport around a closed loop will obviously return the vector to itself. Perhaps, imagine a 2-sphere imbedded in 3-space. The vectors must always be tangent to this 2 sphere. I can move the vectors around, but there is no universal definition of parallel present. See the figure in this article: http://en.wikipedia.org/wiki/Parallel_transport You can see the effects of curvature on the parallel translated vectors there.
Ok, yes, if [X,Y]=0, then the operator makes sense. I feel like your formula can be proved by first showing that [tex]\nabla_XZ=\left.\frac{d}{dt}\right|_{t=0}\tau_{tX}^{-1}Z[/tex] and then using the chain rule on [itex]f(s,t,s,t)=\tau_{sX}^{-1}\tau_{tY}^{-1}\tau_{sX}\tau_{tY}Z[/itex]. (For the first identity, write Z in terms of a parallel frame along an integral curve of X.)
Thanks Matterwave, I am well aware of all that. I've been reading MTW etc, and I think I understand how torsion etc fits in. Like in your diagram, [itex][U,V][/itex] always closes the infinitesimal parallelogram, without any use of parallel propagation (and I found a mathematical proof for it in MTW). I believe the difference with the torsion is that [itex]T(U,V)[/itex] closes the parallelogram formed by the geodesics of [itex]U,V,U\parallel,V\parallel[/itex]. That is, the torsion depends on the choice of connection. (That is obviously true, given the formula). I am unsure if in your picture it would close the inner quadrilateral, as I believe you have to go along geodesics of the given curves. Then again, I'm still not sure. I also wonder if incorporating "twist" into your diagram when there is torsion may help...i.e. [itex]U\parallel[/itex] and [itex]V\parallel[/itex] have components in and out of the page due to their twisting when parallel propagated. What'd I'd like to see is an example of a connection with torsion, and a picture like yours drawn on "graph paper" as such. As for your example of the 2--sphere which is always used to illustrate parallel propagation...well, if, in the usual way, parallel propagation is to "stay at a fixed angle with respect to the coordinates" then isn't a vector parallel propagated around a parallelogram (i.e. defined by latitude A to B, longitude C to D) unchanged? That would imply zero curvature, which is wrong...
If you take, in my picture, U and V to be the tangent vectors to geodesic curves (w.r.t. a metric), then introducing torsion to that picture would mean the parallel propagated vectors can twist into or out of the page, but not in any other direction (or else the connection would no longer be metric-compatible). You can take parallel transport on the 2-sphere to be "move the vector like you would in flat 3-D Euclidean space, and then project the resulting vector onto the tangent space of the sphere". That description matches with the Levi-Civita connection. That is, in fact, what the picture shows, and you can clearly see that the resulting vector is different than the initial vector.
Yes, I also realised they must have assumed the vectors commute. In terms of your suggestion, it is actually [tex] \nabla_XZ=\left.\frac{d}{dt}\right|_{t=0}\tau_{tX}Z [/tex] which I understand. I will think more about solving my problem, but I still don't see how. Cheers
Ah, true, I agree with the first point. As for my comment about it needing to be geodesic, it follows from Visser's calculation that for a parallelogram ABCD, the position reached by going parallel propagating [itex]Y[/itex] along the geodesic of [itex]X[/itex] for [itex]\lambda_1[/itex] and then traveling [itex]\lambda_2[/itex] along the geodesic of this propagated vector, minus the position reached by doing the reverse, is equal to [tex]2{T^a}_{bc}X^b X^c \lambda_1 \lambda_2 + O(\lambda^3). [/tex] I am not sure if taking the limit of small [itex]\lambda[/itex] is the same as covariant derivatives, as you also need to get the geodesic equation in. As for the second point, I've discovered that since lines of constant latitude are not geodesics, a vector will rotate as it is transported along them. Now I just need to convince myself how that works. I do not understand your description however --- do you mean to move the vector along the path on the 2--sphere as you would in 3d Euclidean space? Ah, and then project it...I am going to get a ball and a toothpick to check with that lines of latitude cause precession, but I think I see what you mean. Now, all that remains is the equation...
That's odd. Have you done this computation yourself? Because as far as I can tell, the formula in wiki is wrong by a sign. I.e., I find that [tex]\nabla_{\dot{x}_0}Y=\lim_{t\rightarrow 0}\frac{\tau_{x_t}^{-1}Y_t-Y_0}{t}[/tex] And this is also what my book on riemannian geometry tells me.
But anyway, this detail aside, the idea I was hinting to is the following. Set [itex]f(x_1,x_2,x_3,x_4):=\tau_{x_1X}^{-1}\tau_{x_2Y}^{-1}\tau_{x_3X}\tau_{x_4Y}Z[/itex] Notice that [itex]\tau_{sX}^{-1}=\tau_{-sX}[/itex] and [itex]\tau_{tX}^{-1}=\tau_{-tX}[/itex]. We want to compute d²/dsdt of f(s,t,s,t). So by the chain rule, [tex]\left.\frac{d}{dt}\right|_{t=0}f(s,t,s,t)=\frac{ \partial f }{\partial x_2}(s,0,s,0)+\frac{\partial f}{\partial x_4}(s,0,s,0)=\tau_{-sX}(-\nabla_Y(\tau_{sX}Z))+\nabla_YZ[/tex] and [tex]\left.\frac{d^2}{ds dt}\right|_{t=0}f(s,t,s,t)=\frac{\partial^2 f}{\partial x_1\partial x_2}f(0,0,0,0) + \frac{\partial^2f}{\partial x_3\partial x_2}f(0,0,0,0)+0+0=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ[/tex]
I guess it depends on how you define the parallel propagator. In the (different) notation I was using the numerator was [itex]\Gamma(t \to 0)V_{\gamma{t}} - V_{\gamma(0)}[/itex]. You're right though --- after looking at their notation again, [itex]\tau_{x_t} \leftrightarrow \Gamma(0 \to t)[/itex], so it should be [itex]\tau_{x_t}^{-1}[/itex] Excellent. I was getting confused with how one does [tex]\frac{d}{ds}f(s,t,s,t)[/tex] given there are two [itex]s[/itex] slots. Your solution of letting [itex]x_1=s,x_2=t,x_3=s,x_4=t[/itex] is neat. Thanks a bunch quasar987, that has helped a lot. Now I'll head off and do the additional bit where the vector fields do not commute.