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Covariant derivatives and parallelism for tensors November 3, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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3 comments

Time to continue the story for covariant derivatives and parallelism, and do what I promised yesterday on tensors.

Fix a smooth manifold {M} with a connection {\nabla}. Then parallel translation along a curve {c} beginning at {p} and ending at {q} leads to an isomorphism {\tau_{pq}: T_p(M) \rightarrow T_q(M)}, which depends smoothly on {p,q}. For any {r,s}, we get isomorphisms {\tau^{r,s}_{pq} :T_p(M)^{\otimes r} \otimes T_p(M)^{\vee \otimes s} \rightarrow T_q(M)^{\otimes r} \otimes T_q(M)^{\vee \otimes s} } depending smoothly on {p,q}. (Of course, given an isomorphism {f: M \rightarrow N} of vector spaces, there is an isomorphism {M^* \rightarrow N^*} sending {g \rightarrow g \circ f^{-1}}—the important thing is the inverse.) (more…)

Parallelism determines the connection November 2, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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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 {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…)

Covariant derivatives and parallelism November 1, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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7 comments

[Nobody should read this post without reading the excellent comments below.  It turns out that thinking  more generally (via connections on the pullback bundle) clarifies things. Many thanks to the (anonymous) reader who posted them.  -AM, 5/16]

A couple of days back I covered the definition of a (Koszul) connection. Now I will describe how this gives a way to differentiate vector fields along a curve.

Covariant Derivatives

First of all, here is a minor remark I should have made before. Given a connection {\nabla} and a vector field {Y}, the operation {X \rightarrow \nabla_X Y} is linear in {X} over smooth functions—thus it is a tensor (of type (1,1)), and the value at a point {p} can be defined if {X} is replaced by a tangent vector at {p}. In other words, we get a map {T(M)_p \times \Gamma(TM) \rightarrow T(M)_p}, where {\Gamma(TM)} denotes the space of vector fields. We’re going to need this below. (more…)

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