Is every vector field on real line complete?

Question

This question was left as an exercise in my course of Differential Geometry. I am not able to make much progress on it.

Question: Is every vector field on the real line complete?

Attempt:A vector field X along a curve $\sigma : [a,b] \to M$ is a mapping $X: [a,b]\to T(M)$ which lifts $\sigma$; ie $\pi \circ X=\sigma$. A smooth vector field X on M is called complete if $D_t =m$ where is defined on Page 37 of Frank Warner's book as follows:

But I am not able to think in the case of $D_t=\mathbb{R}$ , that under what conditions the equality should hold.

So, can you please help me with this?

Thanks!

Answer 1

No. One way to approach the problem is to pick a function $x(t)$ that blows up in finite time and construct a smooth vector field on $\Bbb R$ for which $x(t)$ is an integral curve.

Hint For example, we know that $x(t) = \tan t$ has an infinite discontinuity at $t = \frac{\pi}{2}$. How can we write $x'$ as a function of $x$?

Computing gives $x'(t) = \sec^2 t = 1 + \tan^2 t = 1 + x(t)^2$, so $x(t)$ is an integral curve of the smooth vector field $(1 + x^2) \partial_x$ on $\Bbb R$, which is thus not complete.