Integral curves and one-parameter groups of diffeomorphisms

[Fredrik]

I think I understand why a vector field must have a unique set of integral curves, but I don't see why they must define a one-parameter group of diffeomorphisms.

Let X be a vector field on a manifold M, and p a point in M. A smooth curve C through p is said to be an integral curve of X if C(0)=p and

X_{C(t)}=\dot C_{C(t)}

for all t in some interval (-a,a). (I hope it's obvious that what I mean by "dot C" is the tangent vector of C). I'm confused about how this defines a one-parameter group of diffeomorphisms. Supposedly, if we write the integral curve at q as C^q, we can define \phi_t(q)=C^q(t) for all t and q, and now this \phi is a one-parameter group of diffeomorphisms. In particular, this would imply that

\phi_t(\phi_s(p))=\phi_{t+s}(p)

which is equivalent to

C^{C^p(s)}(t)=C^p(t+s)

Let's simplify the notation by calling the integral curve at p C, and the integral curve at C(s) B. The equation becomes

B(t)=C(t+s)

How do we prove that this equation holds?

Also, why does the definition of an integral curve require that X=\dot C holds over an interval rather than just at a single point? I suspect that the answer is that as long as the curve is smooth, it doesn't matter.

 

[quasar987]

Since integrality of a curve is a local thing, we may assume that M=R^n.
To see that the equality
<br /> C^{C^p(s)}(t)=C^p(t+s)<br />
holds, fix s. Then notice that when evaluated at t=0, both sides are equal (to C^p(s)). And, if you differentiate both sides with respect to t, you will see that both side satisfy the differential equation
X(\gamma(t))=\dot{\gamma}(t)
By the fundamental theorem on existence and unicity of the solutions to inital value (Cauchy) problems, it must be that both sides agree for all t where they are defined. Now since s was arbitrary, both sides coincide for all s and t where they are defined.

As to your second question... given a vector field X, we say that a curve is an integral curve of X at p if it passes through p, and if X coincides with the its tangent vector in a nbh of p. The idea that this definition tries to capture is that if X is thought of as a speed field for a particle of unit mass, then an integral curve at p is one that, locally near p, corresponds to the motion of a unit mass particle caught in that speed field. This is why integral curves are also called "flow lines": because if X is thought of as the speed of a fluid on M, then an integral curve at p follows the motion of a particle of this fluids passing through p.

If we only require that X\circ C=\dot C hold at t=0 in the definition of an integral curve, then cleary this would not capture the idea we want for there is usually curves other that the one following the "flow lines" satisfying X\circ C=\dot C

 

[Fredrik]

Thanks quasar. That helped. I think I get it now.