Let $M$ be a non-compact smooth manifold. Suppose we have a nowhere-vanishing smooth vector field X. Is this vector field complete?
I know it is when $M$ is compact. However, I am unsure in the non-compact case. I asked a friend of mine and he is pretty sure that he has seen that is, but he couldn't remember why this was true. Could you please explain why it is or is not true.
Note, a vector field is complete if it's flow is global.
I am trying to use the Uniform Time Lemma to prove it but am current failing. Thanks for the help!
Let $M=\mathbb R$ and $X=(x^2+1)\frac{\partial }{\partial x}$. Then $M$ is complete, $X$ does not vanish, but every solution of $\varphi '=\varphi^2+1$ blows up in finite time, because it's of the form $\varphi(t)=\tan (t-C)$. Therefore $X$ is not complete.