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Parallelism determines the connection
*November 2, 2009*

*Posted by Akhil Mathew in differential geometry, MaBloWriMo.*

Tags: connections, covariant derivatives, MaBloWriMo, parallelism

1 comment so far

Tags: connections, covariant derivatives, MaBloWriMo, parallelism

1 comment so far

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