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.

Next, a curve ${c}$ in the smooth manifold ${M}$ is an immersion ${c: J \rightarrow M}$, where ${J}$ is an interval in ${\mathbb{R}}$. We can talk about a vector field along ${c}$ to be a map ${X: J \rightarrow T(M)}$ such that ${X(t)}$ lies above ${c(t) \in M}$. An example is the derivative ${c'}$.

Now assume ${M}$ is given a connection ${\nabla}$. I claim that there is a unique operator ${D}$ sending vector fields along ${c}$ to vector fields along ${c}$ such that:

• If ${X}$ is a vector field along ${c}$ and ${f: M \rightarrow \mathbb{R}}$, then ${D(fX)(t) = (f \circ c)' X + f D( X)(t)}$.Note that ${c'(t) \in (TM)_{c(t)}}$, by definition.
• If ${X}$ is the restriction of a vector field ${\bar{X}}$ on ${M}$, i.e. ${X(t) = \bar{X}(c(t))}$, then$\displaystyle D(X)(t) = ( \nabla_{c'(t)} \bar{X})(c(t)).$

This operator is called the covariant derivative along ${c}$. It is in fact a generalization of the usual directional derivative of vector fields in multivariable calculus, which occurs when you take the connection on ${\mathbb{R}^n}$ with all Christoffel symbols zero.

The first condition means we can, by multiplying ${X}$ by a cut-off function, assume ${X}$ is supported in some coordinate neighborhood ${U}$ with coordinates ${x^1, \dots, x^n}$. In particular, we may even assume that the image of ${c}$ is contained in ${U}$ by shrinking ${J}$ and using local uniqueness (which we prove below). Moreover, we can assume that ${c}$ is one-to-one by shrinking further.

Now, in the local case, we can write ${c(t) = (c^1(t), \dots, c^n(t))}$, and ${X(t) = \sum_i X^i(t) \partial_i}$, where ${\partial_i = \frac{\partial}{\partial x_i}}$. We can extend ${X}$ to ${\bar{X} = \sum_i \bar{X}^i \partial_i }$. Let the Christoffel symbols of the connections be ${\Gamma^k_{ij}}$. We write write what ${\nabla_{c'(t)} \bar{X} }$ looks like, and dive into the algebra. By linearity

$\displaystyle \nabla_{c'(t)} \bar{X} = \sum_{i,k} c'^i(t) \nabla_{\partial_i} \left( \bar{X}^k \partial_k \right).$

This equals by the derivation-like identity for connections

$\displaystyle \sum_{i,k} c'^i(t) \frac{\partial \bar{X}^k}{\partial x^i} \partial_k + \sum_{i,j,k} c'^i(t) \bar{X}^k \Gamma^j_{ik} \partial_j .$

Shifting the indices, collecting terms, and using that ${X}$ is a restriction of ${\bar{X}}$ gives that if we have such an operator ${D}$, then

$\displaystyle D(X)(t) = \sum_j \left( X'^j(t) + \sum_{i,k} c'^i(t) {X}^k(t) \Gamma^j_{ik}(c(t)) \right) \partial_j.$

So we’re basically out of the woods—this expression depends only on ${X}$. Thus we define ${D(X)}$ this way in local coordinates; it is easily checked that the conditions are satisfied locally, and one pieces together the local covariant derivatives to get the global ones. The fact that patching is legal follows from the uniqueness assertion and a partition of unity argument.

Parallelism

A vector field ${X}$ along the curve ${c}$ is said to be parallel if ${D(X) \equiv 0}$. For instance, in the case of ${\mathbb{R}^n}$ with the usual connection, this means that all the components are constant.

Now fix a curve ${c}$ starting at ${p}$ and ending at ${q}$, with interval ${[0,1]}$.

Proposition 1 Given ${v \in T_p(M)}$, there is a unique parallel vector field ${X}$ along ${c}$ such that ${X(0)=v}$.

Indeed, we may assume that ${c([0,1])}$ is contained in a coordinate neigbhorhood, in which case it follows from the fundamental existence and uniqueness theorem on linear ODEs(!) and the local equation for a connection.

Anyway, this means that we can define a map ${\tau_{pq}: T_p(M) \rightarrow T_q(M)}$ as follows: for ${v}$, choose a curve ${c}$ as above, and then take ${c(1) \in T_q(M)}$. It’s smooth because of the smoothness theorem on ODEs. It’s even linear because if ${X,Y}$ correspond to ${v,w}$, then ${X+Y}$ corresponds to ${v+w}$, etc.

The next result tells us what I have been insisting all along—that connections are about connecting different tangent spaces.

Proposition 2 ${\tau_{pq}}$ is a linear isomorphism.

We just need to check that it’s one to one. This follows because the value of the vector field ${X}$ along ${c}$ at ${1}$ determines its value along ${[0,1]}$, because of the uniqueness theorem on ODEs again.

However, ${\tau_{pq}}$ does depend on the curve ${c}$. I believe the extent to which this dependence holds is measured by the holonomy groups, but I don’t (yet) understand what that’s all about, so I’ll let you read about it elsewhere.