The Unapologetic Mathematician

Mathematics for the interested outsider

Vector Fields on Compact Manifolds are Complete

It turns out that any vector field on a compact manifold is complete. That is, starting at any point we can follow the vector field on construct its integral curve as far forward or as far backward as we want. I’ll show this two ways.

First of all, let’s say that there is some open interval I containing a closed interval [-\delta,\delta] such that I\times M is contained in the W from the maximal flow of X. That is, assume that no matter where we start we can flow forward or backward by \delta. But if we start at p_0 this means we can flow forward by \delta to reach a point

\displaystyle p_1=\Phi_\delta(p_0)=\Phi(\delta,p_0)

But then by assumption we can flow forward again by \delta to reach

\displaystyle p_2=\Phi_\delta(p_1)=\Phi_\delta(\Phi_\delta(p_0))=\Phi_{2\delta}(p_0)

And we can keep flowing forward to reach p_n=\Phi_{n\delta}(p_0) for arbitrarily large n. Similarly we can flow backward as far as we want.

So, is there such an interval? In general there doesn’t have to be, but if M is compact there is. Indeed, at each point p we can define the interval I_p=\left(-a(p),b(p)\right) as the maximal open interval on which the integral curve based at p is defined. Here, each a(p) and b(p) is a positive real number or \infty. Since M is compact, each of these functions has a minimum — a and b. And then the interval (a,b) is contained in each I_p, as asserted.

A little more analytically, let \phi:[\alpha,\beta)\to M be an integral curve of a vector field X, and suppose that there is a sequence t_n increasing to \beta for which \lim\phi(t_n)\to p. Then it can only make sense to define \bar{\phi}:[\alpha,\beta]\to M by defining \bar{\phi}(t)=\phi(t) for \alpha\leq<\beta and \bar{\phi}(\beta)=p, for continuity. But now if c:I\to M is the maximal integral curve of X with c(\beta)=p then uniqueness tells us that c(t)=\bar{\phi}(t) for \alpha\leq t\leq\beta.

So what happens when M is compact? In this case, any sequence of t_n converging to \beta gives rise to a sequence of \phi(t_n) converging to some common p. Thus we can always extend any integral curve on a right-open interval [\alpha,\beta) to the closed interval [\alpha,\beta], and then past \beta to a somewhat longer right-open interval, and so on as far as we want to go. And thus, again, every integral curve can be extended forever in either direction, which makes X complete.


June 1, 2011 Posted by | Differential Topology, Topology | 1 Comment