Parallelism determines the connection

I’m going to try participating in Charles Siegel’s MaBloWriMo project of writing a short post a day for a month.  In particular, I’m categorizing yesterday’s post that way too.  I’m making no promises about meeting that every day, but much of the material I talk about lends itself to bite-sized pieces anyway.

There is a nice way to tie together (dare I say connect?) the material yesterday on parallelism with the axiomatic scheme for a Koszul connection. In particular, it shows that connections can be recovered from parallelism.

So, let’s pick a nonzero tangent vector , where is a smooth manifold endowed with a connection , and a vector field . Then makes sense from the axiomatic definition. We want to make this look more like a normal derivative.

Now choose a curve with . Then I claim that

This can be proved by a direct calculation in local coordinates, but here is a slightly more devious argument. By shrinking, assume is an embedding and not just an immersion. Now let be a basis for the tangent space , and consider the parallel vector fields on the curve with ; extend them to vector fields on some neighborhood of with the same notation. Then, if necessary by shrinking the neighborhood of definition, we can assume the are linearly independent, and by assumption of parallelism.

We can write for some smooth . Then

and by parallelism

Note that parallel translation is linear, so . Summing, we’re done.

Now thus parallel translation induces isomorphisms on the tangent spaces. In particular, we can consider the tensor bundles and find isomorphisms between the fibers at any two points depending upon a curve between them. Then we can use a limit definition to define a connection on tensor fields too. But I will come back to this tomorrow.