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 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.
Parallelism
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.
November 4, 2009 at 10:31 pm
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