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covariant derivative,torsion, curvature, torsionless, flat, Christoffel symbol
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Great text on connections : short, concise and extremely readable.

I think these references (as you've noted) by Elie Cartan could be mentionned :

1- "Leçon sur la géométrie des espaces de Rieman", E.Cartan. Gauthier-Villars, Paris, 1963.
This is lecture given by Cartan in 1925.

2- "Sur les variétés à connexion affine et la théorie de la relativité généralisée", E. Cartan, Annales scientifiques de l'ENS, 3ème série, tome 40 (1923), 325-412.
This article (and the following two) are available on line on "Numdam":
or directly at
This is an extremely beautifull article.


Just wanted to say thanks! Numdam looks to be a fantastic resource. I will certainly take up your advice. Best wishes, RM

I think you are definitely on the right track with your comments! Cartan's idea about what a connection is is nowadays formalized as a "Cartan connection". Let me tack a stab at an informal explanation. But first a disclaimer: I can't pretend to be an expert on this. I've noticed the discrepancy, pondered this question, tried to talk to people about it, but the following interpretation is strictly my own synthesis based on reading Cartan. I have some confidence in it though, because Sharpe talks about similar ideas, albeit in terms of a different metaphor: "a meteor streaking through the principal bundle". It appears to be the same idea though.

The modern idea of connection is that it is the derivative of the parallel transport operation. So what is parallel transport, appropriately generalized and abstracted?

We start with a fibre bundle E->B and say that "parallel transport" is a structure that gives us a unique way of lifting paths on B to paths on E. The motivation for this comes about when E is the the bundle of linear frames. We have a path on the base manifold B, and when we lift it to E, we get an isomorphism between the space of frames at the start of the path and the space of frames at the end of the path. To get this kind of lifting we split the tangent bundle of E. We already have the naturally Vertical bundle VE, but now to add a connection we add a horizontal complement HE. The choice of HE is the connection.

The connection (which can also be described as an injection of TB into TE that splits the exact sequence VE->TE->TB) maps every tangent vector in TB uniquely to a tangent vector in TE. Now when we slide along a path on B, the connection maps the velocity in TB to a velocity in TE, and voila, we get a path in TE. Usually, we have some kind of Lie group acting on the fibres so we also ask that the splitting TE=VE+HE be equivariant with this action. That correspond to the equivariance of the parallel transport with respect to the group action. Makes perfect sense. If we parallel transport a certain frame Fx, from point x\in B to a frame Fy at point y\in B along a fixed path, we should have enough information to parallel transport every other frame as well. Thus if g is any linear transformation, g*Fx should be parallel transported to g*Fy along the fixed path in question. To make this work infinitesimally, we need to make sure that the pushforward g* sends HE to HE.

OK, that's a sketch of the modern idea. Now Cartan thought differently. Let's start with Riemannian structure. Let's say you are encased in a ridid body (your vehicle) at some point of a Riemannian manifold. To be concrete let's say we are on a surface embedded in 3d Euclidean space. To move around on the surfaces one needs to exert 2 different kinds of forces: an external force directed along the normal to the surface which keeps you trapped in the surface, and an internal force that changes your speed and heading (motion with purely external accelration = geodesic motion). However as one moves around, the orientation of one's vehicle is also undergoing changes. However, this change is only perceptible in the ambient 3D space. If one is a 2d creature trapped in Flatland, one cannot perceive the ambient 3D, and there is no way to compare the tangent spaces at different points. Thus, a 2D flatland being cannot internally perceive the rigid body rotation that one's vehicle undergoes as it moves around on the surface.

However, there is something that even a 2d flatlander can do and measure. Starting at a fixed point, one can take a sheet of paper (Euclidean plane) mark off an origin to represent ones current location, draw a compass rosette, and associate the directions on paper to actual physical directions in the surface. Then one can start driving/flying by choosing an acceleration relative to ones frame of reference. In the next instant one is somewhere else, but still the orientation of one's rigid body vehicle gives you a point of reference and you can keep on steering. Basically, you can draw a smooth path in your navigational chart (the Euclidean plane) and use that path to drive your vehicle around on the surface.

Now to do meaningful measurement in this setting one has to drive back back to one's starting point, and this is where the first miracle happens. If one draws a closed loop in ones navigational chart and uses this loop to steer by around the surface, one will find that one returns back to ones starting point. This is called absence of torsion! However, when one returns back to one's point of origin, one will find that the orientation of the vehicle upon arrival is different from the orientation at the time of departure. THis is due to the presence of curvature. The change in orientation is called holonomy.

So what Cartan did was to abstract this. To him a connection was a mathematical structure that allowed one to translate paths from a navigational chart, (Euclidean space in case of Riemannian structure, affine space in the case of affine structre, the sphere in the case of conformal structure, Projective space in the case of projective structure, etc) to the underlying manifold. Now for Euclidean and affine structure, this ends up being the same story as the standard parallel transport version of connections. However, when one's navigational chart is something like Projective space, then parallel transport won't cut it and you need some other kind of structure. I won't go into the technicalities: these can be had from Sharpe's book and also from Kobayashi's "Transformation groups".

A good question along these lines is "what is the difference between an affine connection and a linear connection"? The answer, as far as I can tell, is that these are two different ways to encode the same information. With an affine connection we bundle torsion and connection into one big (n+1)x(n+1) matrix of 1-forms. For linear connections, we have just the nxn connection matrix of 1-forms. However, this connection matrix lives on the bundle of linear frames, where we have the R^n valued canonical 1-form (the \theta^i as they are usually named) and these end up being exactly the extra bits from the Cartan connection. So the answer seems to be that "affine connection" = "linear connection", but we formalize/encode the same information differently.

Feedback would be appreciated.

Sorry, it's taken me so long to reply. Your latest comment is steering the discussion in a direction with which I am not very familiar: constrained Lagrangian systems and non-holonomic connections.

It seems to me that one could define a "connection" without any kinds of equivariance built in. One could start with a fibre bundle and define the horizontal subspace by splitting the exact sequence VE->TE->TB.
So, for example, we could get a connection on the tangent space TM of a manifold. I have not really encountered a use for a non-equivariant connection, but maybe in mechanics this is exactly what is needed?

You give the example of B=J1(R,R) (first jets of maps from R to R)
This is indeed, 3-dimensional, with a non-holonomic distribution, which we may consider to be the horizontal complement. So we can raise paths from R (the base) to B (this is usually called prolongation). How can we think of this as an affine connection though? The affine group (2 dimensional in this case, dilations+translations) acts on the 2d fibres of B->R. The fibres contain the zeroth and the first jet info: translations change the zeroth order part and the dilations the first order part.

To get a connection we need an aff(2)-valued 1-form on B=J1. Now I am stuck, because I don't see how to do this. To get a Cartan connection, aka an absolute parallelism we need a 3d group that contains aff(2), because dim B has to equal dim G. So I didn't quite follow your example (sorry for the rambling remarks, I was trying to figure things out).

In Kobayashi's "Transformations groups" he talks about a natural construction that lets one "thicken" the fibre bundle and turn a Cartan connection into an "ordinary" connection. Would this construction be helpful in your exposition? I thought that curvature=infinitesimal holonomy. You seem to be suggesting that torsion can also manifest as holonomy. Perhaps when one does Kobayashi's "thickening construction" that is exactly how torsion will manifest?

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