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…)

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…)

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…)