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


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 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


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