The Unapologetic Mathematician

Mathematics for the interested outsider

Integral Curves and Local Flows

Let \mathfrak{X}M is a vector field on the manifold M and let q be any point in M. Then I say there exists a neighborhood V\subseteq M of q, an interval I\subseteq\mathbb{R} around 0, and a differentiable map \Phi:I\times V\to M such that

\displaystyle\begin{aligned}\Phi(0,p)&=p\\\Phi_*\left(\frac{\partial}{\partial t}(t,p)\right)&=X\left(\Phi(t,p)\right)\end{aligned}

for all t\in I and p\in V. 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 \frac{\partial}{\partial t}(t,p) is the inclusion \iota_{p*}\left(\frac{d}{dt}(t)\right) of the canonical vector \frac{d}{dt}(t)\in\mathcal{T}_t\mathbb{R} which points in the direction of increasing t. That is, \iota_p:I\to I\times V includes the interval I into I\times V “at the point p\in V“, and thus its derivative carries along its tangent bundle. At each point of an (oriented) interval I there’s a canonical vector, and \frac{\partial}{\partial t}(t,p) is the image of that vector.

Further, take note that we can write the left side of our second condition as

\displaystyle\Phi_*\left(\iota_{p*}\left(\frac{d}{dt}(t)\right)\right)

The chain rule lets us combine these two outer derivatives into one:

\displaystyle\left[\Phi\circ\iota_p\right]_*\left(\frac{d}{dt}(t)\right)

But this is exactly how we defined the derivative of a curve! That is, we can write down a function c=\Phi\circ\iota_p:I\to M which satisfies c'(t)=X(c(t)) for every t\in I. We call such a curve an “integral curve” of the vector field X, and when they’re collected together as in \Phi we call it a “local flow” of X.

So how do we prove this? We just take local coordinates and use our good old existence theorem! Indeed, if (U,x) is a coordinate patch around q then we can set G=x(U), a=x(q), and

\displaystyle F=(X^1,\dots,X^n)\circ x^{-1}:G\to\mathbb{R}^n

where the X^i are the components Xx^i of X relative to the given local coordinates.

Now our existence theorem tells us there is a neighborhood W\subseteq G of a, an interval I around 0, and a map \psi:I\times W\to G satisfying the conditions for a flow. Setting V=x^{-1}(W) and \Phi(t,p)=x^{-1}\left(\psi(t,x(p))\right) we find our local flow.

We can also do the same thing with our uniqueness theorem: if c and \tilde{c} are two integral curves of X defined on the same interval I, and if c(t_0)=\tilde{c}(t_0) for some t_0\in I, then c=\tilde{c}.

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.

May 28, 2011 Posted by | Differential Topology, Topology | 4 Comments

   

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