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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?

2 Answers 2

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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?2012-07-01
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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.

  • 0
    could you explain me the first one?2012-07-01