2
$\begingroup$

Given a smooth vector field on a closed smooth manifold, does the flow of that vector field exists for all time $t \in R$? This may be a well-known and elementary conclusion, but I just wnat to confirm it to me.

  • 2
    If "closed" means compact, then yes.2010-10-23
  • 0
    If "closed" does not mean compact, then no. $\dot x=x^2$2010-10-23
  • 0
    Usually *closed*, as applied to manifolds, means *compact and without boundary*. (On a compact manifold with boundary it is clear that solutions need tobe defined on all of $\mathbb R$: consider for example on the manifold $[0,1]\times S^2$ the vector field $d/dr$ with hopefully self-evident notation)2010-10-23

1 Answers 1

5

The local existence theorem for ODEs gives a lower bound for the domain of existence of solutions in terms of a Lipschitz constant. On a compact manifold $M$, you can find a global Lipschitz constant, and then the theorem gives you an $\epsilon>0$ such that for every point $x\in M$, the solution that starts at $x$ is defined at least in $(-\epsilon,\epsilon)$. With this you immediately get existence for all values of $t$.