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Parallelism determines the connection November 2, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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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 {Y \in T_p(M)}, where {M} is a smooth manifold endowed with a connection {\nabla}, and a vector field {X}. Then {\nabla_Y X \in T_p(M)} makes sense from the axiomatic definition. We want to make this look more like a normal derivative.

Now choose a curve {c: (-1,1) \rightarrow M} with {c(0)=p,c'(0) = Y}. Then I claim that

\displaystyle \nabla_Y X = \lim_{s \rightarrow 0} \frac{ \tau_{p, c(s)}^{-1} X(c(s)) - X(p) }{s}. (more…)