Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 ?

share|cite|improve this question
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

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} $

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.