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 ? 

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} $
