November 3, 2009

Covariant derivatives and parallelism for tensors

Posted by Akhil Mathew under differential geometry, MaBloWriMo | Tags: , , , , |
[2] 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…)

 

November 2, 2009

Parallelism determines the connection

Posted by Akhil Mathew under differential geometry, MaBloWriMo | Tags: , , , |
1 Comment 

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…)

 

November 1, 2009

Covariant derivatives and parallelism

Posted by Akhil Mathew under differential geometry, MaBloWriMo | Tags: , , , |
1 Comment 

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…)