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Could any one help me to solve these two problems on vector fields

  1. Any $\mathbb{C}^{\infty}$ vector field on a compact manifold is complete.

  2. Is every vector field on $\mathbb{R}$ complete?

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2 Answers 2

up vote 3 down vote accepted

The 1st is true, not the 2nd. The flow associated to $\dot{x}=x^2$ is given by $\phi_t(x)=x/(1-tx)$, and obviously does not exist for every $t$.

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could you explain me the first one? –  Une Femme Douce Jul 1 '12 at 22:04

For the first, use the neighbourhood definition of compactness and do a proof by contradiction. The second is not true and you only have to find a suitable counter example.

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could you explain me the first one? –  Une Femme Douce Jul 1 '12 at 22:04

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