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I met a statement saying:

The product structure $V=M\times\mathbb R$, with $M$ an orientable $3$-manifold, implies the existence of a global coframe (4 globally defined linearly independent 1-forms). The property does not extend to higher dimensions.

I can not prove this and do not know why this dimension 3 is special. Can someone give a proof and counterexamples?

Thank you.

Yikun Qiao
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    Every 3 orientable manifold is parallelizable (see here). I am not sure if the product $M \times \mathbb R$ will make an easier argument though. –  Dec 18 '17 at 18:52
  • Certainly the analogous result is true for any $X\times\Bbb R^k$, whenever $X$ is parallelizable. So I don't know what "the property does not extend to higher dimensions" means. But notice, also, that $S^2$ is not parallelizable and yet $S^2\times\Bbb R$ is. – Ted Shifrin Dec 20 '17 at 00:13

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