Parallel Transport – How to Explain Parallel Transport to Undergraduates

dg.differential-geometryintuitionmathematics-educationmg.metric-geometryriemannian-geometry

The title is a bit deceiving, because what I really mean is the parallel transport that corresponds to the Levi–Civita connection.

This is in the vein of many other questions on mathoverflow:

But the focus is different.

Let me summarize my understanding given answers in the questions above:

(Very Rough) Summary of Answers in Previous Questions

  1. There exists an interpretation in terms of $G$-structures, as in the chosen answer (by Chris Schommer-Pries) to What is torsion in differential geometry intuitively?.
  2. There exists an interpretation as having some universal property, as described in the top answer (by Robert Bryant) to What is the Levi-Civita connection trying to describe?.
  3. There exists a deceiving but appealing interpretation by parallelograms whose sides don't really lie in the same space, as in the answer by Gabe K to What is the Levi-Civita connection trying to describe?. (Gabe K did a heroic effort to make sense of the nonsensical diagram, and I thank him dearly.)
  4. There exists an interpretation regarding rolling the shape on a surface (Rolling without slipping interpretation of torsion).

But ultimately, none of that is something that I can intuitively sell to an undergraduate, and by undergraduate I really mean my heart. In the bottom of my heart, I need a better explanation, one that starts with desirable properties, and then proceeds through existence and uniqueness.

Outline of the Type of Intuition I Desire

I want to start with some desirable behaviors, which I allow to be external (i.e, to reference a given embedding of the Riemannian manifold into $\mathbb{R}^n$), and then say that the only notion of parallel transport that satisfies these conditions must be the Levi–Civita connection. (Any reasonable notion of parallel transport will respect the metric, so I'm really thinking of the torsion-free condition.)

A base case of a desirable condition is that for the Riemannian manifold $\mathbb{R}^n$, parallel transport is the trivial thing. (If one identifies the tangent bundle with $\mathbb{R}^n\times\mathbb{R}^n$ then for any path $\gamma$ the parallel transport of the tangent vector $(\gamma(0),v)$ at $\gamma(0)$ to $\gamma(1)$ via $\gamma$ is the tangent vector $(\gamma(1),v)$ at $\gamma(1)$.)

Next, we would like some way to generalize to a general Riemannian manifold. Let $(M,g)$ be a Riemannian manifold, and let $p\in M$ be a point. Then by the implicit function theorem we can have a chart $f:V\rightarrow U\subset \mathbb{R}^d$ where $0\in V\subset \mathbb{R}^n$, and $p\in U\subset M$, such that $f(0)=p$ and such that $f$ is the identity on the first $n$ coordinates.

My next thought is to look at the most intuitive case of torsion-freeness, which is the case of commuting fields $X$ and $Y$. By change of coordinates, we can assume WLOG that on $V$ the vector fields $X$ and $Y$ are defined via the constant functions $X(v)=e_1$ and $Y(v)=e_2$. One can then express $X$ on $Y$ on $M$ via the derivative of $f$.

But I'm missing multiple components to proceed.

So let me ask this in terms of several more explicit questions.

Questions

  1. If a connection satisfies that $\nabla_XY=\nabla_YX$ for any commuting set of vector fields $X$ and $Y$, then is it torsion-free? (In other words, if you're torsion-free on commuting vector fields, are you torsion-free for all vector fields?)
  2. What intuitive desirable condition (that is allowed to use a given embedding of $M$ into some $\mathbb{R}^d$), combined with, or perhaps generalizing the desired behavior of parallel transport on Euclidean space, would uniquely determine it as satisfying $\nabla_XY=\nabla_YX$ for commuting vector fields? (Perhaps something about geodesics? Or volumes? I don't really know what's the missing component here.)
  3. I feel like Ben McKay's answer to What is the Levi-Civita connection trying to describe? is coming close to what I want, but I did not get to the bottom of it. It appeared at first that he was saying that the Levi–Civita parallel transport is simply parallel transporting in the ambient space, and then projecting to the tangent plane. But in retrospect, my interpretation is clearly wrong. (Imagine for example an upward pointing vector on the equator of a sphere, being parallel transported to the top. If you parallel transport in $\mathbb{R}^3$ you'll get a vector pointing up, which projected to the tangent space will be the $0$ vector.)
  4. A little more vaguely, in case you have an entirely different notion in mind, how would you explain parallel transport to the undergraduate in your heart?

Best Answer

This may not reallly be an answer that you like, but I think that, maybe you misunderstood what Ben McKay was trying to describe. Here is a more explicit, extrinsic description that may help:

Suppose that $M^m\subset\mathbb{E}^n$ is an isometrically embedded submanifold of Euclidean $n$-space. Let $\gamma:(a,b)\to M^m$ be a smooth curve in $M$ and let $v:(a,b)\to\mathbb{E}^n$ be a curve of vectors along $\gamma$, i.e., $v(t)$ lies in the tangent space $T_{\gamma(t)}M$ for all $t\in (a,b)$. Say that $v$ is parallel (along $\gamma$) if $v':(a,b)\to\mathbb{E}^n$ is normal to $TM$ along $\gamma$, i.e., $v'(t)\perp T_{\gamma(t)}M$ for all $t\in(a,b)$. In other words, the velocity of $v$ is always perpendicular to the tangent vectors to $M$ at the point of tangency.

Then the (easily proved) proposition is that this notion of a tangent vector field along a curve being parallel along $\gamma$ does not depend on the choice of the isometric embedding, i.e., it is intrinsic to the metric induced on $M$ by its embedding. More generally, if $v:(a,b)\to\mathbb{E}^n$ is tangent along $\gamma$, then letting $D_\gamma v(t)$ be the orthogonal projection of $v'(t)$ onto $T_{\gamma(t)}M$ yields another curve $D_\gamma v:(a,b)\to\mathbb{E}^n$ that is tangent along $\gamma$, and this operation (actually a derivation) on tangent fields along $\gamma$ depends only on the induced metric on $M$. Since it is independent of the choice of isometric embedding, it is the 'covariant part' of the ambient derivative, i.e., the 'covariant derivative'.

For example, it follows from the definition that if $v$ is a parallel tangent vector field along $\gamma$, then the length of $v$ is constant. Then the existence and uniqueness of 'parallel transport' follow by elementary ODE arguments. The Leibnitz rule for the 'covariant derivative' and other properties are easily derived from the definition as well.

Once you know that $\nabla_{\gamma'}v$ for a curve of tangent vectors depends only on the metric, it's natural to want to find a formula for it that uses only on the metric and not the (superfluous) isometric embedding. That is what leads to the usual characterizations.

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