Covariant derivatives and parallelism November 1, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: connections, covariant derivatives, ordinary differential equations, parallelism
[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.
First of all, here is a minor remark I should have made before. Given a connection and a vector field , the operation is linear in over smooth functions—thus it is a tensor (of type (1,1)), and the value at a point can be defined if is replaced by a tangent vector at . In other words, we get a map , where denotes the space of vector fields. We’re going to need this below.
Next, a curve in the smooth manifold is an immersion , where is an interval in . We can talk about a vector field along to be a map such that lies above . An example is the derivative .
Now assume is given a connection . I claim that there is a unique operator sending vector fields along to vector fields along such that:
- If is a vector field along and , then .Note that , by definition.
- If is the restriction of a vector field on , i.e. , then
This operator is called the covariant derivative along . 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 with all Christoffel symbols zero.
The first condition means we can, by multiplying by a cut-off function, assume is supported in some coordinate neighborhood with coordinates . In particular, we may even assume that the image of is contained in by shrinking and using local uniqueness (which we prove below). Moreover, we can assume that is one-to-one by shrinking further.
Now, in the local case, we can write , and , where . We can extend to . Let the Christoffel symbols of the connections be . We write write what looks like, and dive into the algebra. By linearity
This equals by the derivation-like identity for connections
Shifting the indices, collecting terms, and using that is a restriction of gives that if we have such an operator , then
So we’re basically out of the woods—this expression depends only on . Thus we define 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.
A vector field along the curve is said to be parallel if . For instance, in the case of with the usual connection, this means that all the components are constant.
Now fix a curve starting at and ending at , with interval .
Proposition 1 Given , there is a unique parallel vector field along such that .
Indeed, we may assume that 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 as follows: for , choose a curve as above, and then take . It’s smooth because of the smoothness theorem on ODEs. It’s even linear because if correspond to , then corresponds to , etc.
The next result tells us what I have been insisting all along—that connections are about connecting different tangent spaces.
Proposition 2 is a linear isomorphism.
We just need to check that it’s one to one. This follows because the value of the vector field along at determines its value along , because of the uniqueness theorem on ODEs again.
However, does depend on the curve . 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.