Parallelisable three dimensional orientable manifolds and Stiefel's theorem On Steenrod's book "The topology of fibre bundles" by the end of page 203 and beginning of page 204, Steenrod claims that any orientable three dimensional manifold is parallelisable, and he cites an article by Stielfel: "Richtungsfelder und Fernparallelismus in n-dimensionalen Mannigfaltigkeiten".
I cannot read mathematics in German; thus my question is:

Does Stielfel's theorem about the parallelisability of orientable three manifolds applies to noncompact ones, or his proof relies on the assumption that the manifold is compact? 

If his proof depends on the manifold being compact,

Is it true that any noncompact and orientable three dimensional manifold is parallelisable?     

PVAL suggested a proof in his answer to this question: Elementary proof of the fact that any orientable 3-manifold is parallelizable . However, I should like to see a reference in the literature with more details than Steenrod's comments (in English, if it is the case that Stielfel's theorem holds for noncompact manifolds).

If you just want to deduce the noncompact case from the closed one, this requires little machinery.
Basically, you find first an exhaustion of $M$ with connected compact manifolds with boundary $M_k$. Then you inductively construct linearly independent vector fields $X,Y,Z$ on each $M_k$. Extending them from $M_k$ to $M_{k+1}$ is not trivial, and can be done e.g. taking an appropriate harmonic extension. See my answer to a very similar question here.
