A More on Parallel Transport: Existence & Uniqueness

[pervect]

The recent thread on parallel transport has raised a couple of things I'd like to review for my own sake. I'll address them one at a time as my time permits.

The first question is this. If we offer $t^a \nabla_a u^b$ or the equivalent $\nabla_{\vec{t}} u^b$ as the definition of parallel transport of a vector $\vec{u}$ along some curve C with tangent $\vec{t}$, how do we show the existence and uniqueness of parallel transport? To be a bit more clear, we have some curve C with a tangent $t^a$, and we are assuming that some derivative operator $\tilde{\nabla}_a$ exists and is well-defined. At this point we aren't making any more specific assumptions about $\tilde{\nabla}_a$, in fact if we follow Wald's logic we are assuming that there are many possible ways to define a derivative operator that meet the necessary axioms, and we are free to pick any of them. Eventually, we'll realize that these other derivative operators yield other connections, and that for the purposes of doing GR the connection we are interested in is the Levi-Civita connection. But at this point we are only assuming that we've singled out one specific possibility for the derivative operator, and we want to show that this implies we've also singled out some specific notion of parallel transport.

Given then, that we have a well-defined derivative operator, if we have a curve, and we have a vector on the curve, how do we go about showing that this definition yields a unique answer to the question of parallel transporting said vector along said curve? I think in the recent thread, there were some concerns about the existence and uniqueness of this concept of parallel transport. Having a definition, if it's a good one, should address these concerns.

 

[andrewkirk]

John Lee, in his book Riemannian Manifolds presents a proof of this as Theorem 4.11 ('Parallel Translation'). The derivative used to determine the parallellness of the transported vector field is a general covariant derivative, not required to be the Levi-Civita.

It is based on a theorem about the existence and uniqueness of a solution to a certain type of linear ODE, which is presented by Lee as Theorem 4.12.

I google searched some words from the hard copy and came up with this link to the relevant pages on Google books. Maybe it will work for others.

 

[martinbn]

The question was answered but I want to make a side comment. You start with a curve and a vector field along the curve and you want to be able to tell whether it is parallel along the curve. For that you have to be able to differentiate it in the direction of the tangent vector. And there is a subtle point here. If you have a vector field on the manifold you can restrict it to the curve and have a field along the curve. But not every field along the curve is of that form. For example when the curve self intersects you cannot always extend a field on the curve to a field on the whole manifold. So in general you cannot just extend and differentiate. So you have to prove (as done in diff.geom. books, I believe in Lee's as well, and some GR books) that there is a derivative operator that acts on fields on the curve that agrees with the derivative operator from the connection when the field is extendable.