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I know that every vector field on a compact manifold is complete. The question of whether every non-compact manifold admits an incomplete vector field seems to follow naturally. I'd hazard a guess that the answer is no, but I can't think of an appropriate non-compact space to use as an example. Could someone give me a hint as to how to think of the right space, or inform me that my intuition is wrong? Many thanks in advance!

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I think the answer is yes. A non-compact manifold $M$ should admit a proper embedding $f: [0,\infty)\to M$. Push forward an incomplete field from $[0,\infty)$ to $M$, and extend to $M$ in an arbitrary way. –  user53153 Dec 20 '12 at 17:54
    
Thanks - that's a convincing argument. My intuition was clearly quite a way off! –  Edward Hughes Dec 20 '12 at 21:37

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