Integral Curves and Local Flows
Let is a vector field on the manifold and let be any point in . Then I say there exists a neighborhood of , an interval around , and a differentiable map such that
for all and . These should look familiar, since they’re very similar to the conditions we wrote down for the flow of a differential equation.
It might help a bit to clarify that is the inclusion of the canonical vector which points in the direction of increasing . That is, includes the interval into “at the point “, and thus its derivative carries along its tangent bundle. At each point of an (oriented) interval there’s a canonical vector, and is the image of that vector.
Further, take note that we can write the left side of our second condition as
The chain rule lets us combine these two outer derivatives into one:
But this is exactly how we defined the derivative of a curve! That is, we can write down a function which satisfies for every . We call such a curve an “integral curve” of the vector field , and when they’re collected together as in we call it a “local flow” of .
So how do we prove this? We just take local coordinates and use our good old existence theorem! Indeed, if is a coordinate patch around then we can set , , and
where the are the components of relative to the given local coordinates.
Now our existence theorem tells us there is a neighborhood of , an interval around , and a map satisfying the conditions for a flow. Setting and we find our local flow.
We can also do the same thing with our uniqueness theorem: if and are two integral curves of defined on the same interval , and if for some , then .
Thus we find the geometric meaning of that messy foray into analysis: a smooth vector field has a smooth local flow around every point, and integral curves of vector fields are unique.