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}.

This can be proved by a direct calculation in local coordinates, but here is a slightly more devious argument. By shrinking, assume {c} is an embedding and not just an immersion. Now let {v_1, \dots, v_n} be a basis for the tangent space {T_p(M)}, and consider the parallel vector fields {X_1, \dots, X_n} on the curve {c} with {X_i(0) = v_i}; extend them to vector fields on some neighborhood of {c((-1,1))} with the same notation. Then, if necessary by shrinking the neighborhood of definition, we can assume the {X_i} are linearly independent, and {\nabla_Y X_i = 0} by assumption of parallelism.

We can write {X = \sum f_i X_i} for some smooth {f_i}. Then

\displaystyle \nabla_Y X = \sum (Yf_i)X_i (p)

and by parallelism

\displaystyle \lim_{s \rightarrow 0} \frac{ \tau_{p, c(s)}^{-1} (fX_i)(c(s)) - X_i(p) }{s} = X_i \lim_{s \rightarrow 0} \frac{ f(c(s)) - f(p)}{s} = (Yf_i) X_i.

Note that parallel translation is linear, so {\tau_{p, c(s)}^{-1} (fX_i)(c(s)) = f(c(s)) X_i(p)}. Summing, we’re done.

Now thus parallel translation induces isomorphisms on the tangent spaces. In particular, we can consider the tensor bundles {T^{r.s}(M)} and find isomorphisms between the fibers at any two points {p,q} 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.