"Completing" a vector field on a non-compact manifold $M$

Question

Suppose I have a non-compact smooth manifold $M$ and an arbitrary nowhere-vanishing smooth vector field $X$ on $M$ which is not complete.

Is there a way to create a smooth vector field $V$ that is still nowhere vanishing but now complete. For instance, maybe we can find a smooth function $f$ such that $Y=fX$ fills my requirements.

Thanks for the help!

Notice that there is no need to suppose the manifold is not compact, as that is implied by the existence of a non-complete vector field :-) — Mariano Suárez-Álvarez, May 02 '13 at 00:17

score 13 · Accepted Answer · edited Apr 13 '17 at 12:58

Pick any complete Riemannian metric on your manifold (such things exist; see this) and normalize the vector field —you can do this because the vector field does not vanish anywhere.

Pick a point $p$ and let $(a,b)$ be the maximal domain of the integral curve $\gamma$ of the resulting vector field which starts at $p$ at time $0$. If $b<\infty$, then $\gamma(t)$ must leave every compact set as $t\to b$, so in particular it leaves every closed ball centered at $p$ as $t\to b$ (closed balls are metrically compact in a geodesically complete manifold: this is one form of the Hopf–Rinow theorem; see this for a statement) Since the curve is parametrized by arc length, this is impossible.

A similar argument shows that $a$ must be $-\infty$, so the normalized vector field is complete.

"Completing" a vector field on a non-compact manifold $M$

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