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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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    Thanks - that's a convincing argument. My intuition was clearly quite a way o$f$f!2012-12-20

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