a theorem in topology

Is there anyone know there is a theorem in topology which states that a compact manifold "parallelizable" with N smooth independent vector fields. must be an N-torus? and why the vector field here is parallel to manifold ?

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The general-topology tag is for questions relating only to topological structure, not manifold or differentiable structure (see the tag wiki). – joriki Jan 23 '13 at 21:52
What do you mean with parallel to [the] manifold? The definition of parallelizability for a n-manifold is exactly, that there are n (smooth) vector fields which provide a basis for each tangent space. This is equivalent to what we call triviality of the tangent bundle and is fulfilled for some non-tori too. Take the three dimensional sphere for example. – Ben Jan 23 '13 at 22:18

1 Answer

I think you are talking about a theorem due to V.I. Arnold: you can find more details in "Mathematical methods of classical mechanics", chapter 10. Here is the statement.

Theorem: Let $M$ be a n-dimensional compact and connected manifold and let $Y_{1},...,Y_{n}$ be smooth vector fields on M, commuting each other. If, for each $x \in M$ $(Y_{1}(x),...,Y_{n}(x))$ is a basis of the tangent space to M at x, then M is diffeomorphic to $\mathbf{T}^{n}$

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