What are interesting corollaries of a manifold being parallellizable? This is a heavily edited (in fact, a complete rewrite) of a question I asked badly a few days ago. I am editing as opposed to asking a new question as there are already several relevant answers.
I thought that perhaps a parallelizable manifold could be said to be "simpler" in some way.
Initially I thought things like "maybe it is possible to equip them with a flat metric" however, this is obviously false as $S^3$ is parallelizable. Or "maybe it is always possible to embed them in $\mathbb{R}^n$" for some small where $n$ smaller than twice the dimension of the manifold. Or perhaps "there is a structure on these manifolds that is not present on normal ones, like a Lie group structure but obviously not that". Or perhaps "we can say something about the topological invariants of such a manifolds"?
In particular, I was wondering if there is any intuitive or deep reason behind the statement "all orientable 3-manifolds are parallellizable". Does it have anything to do with the 3-sphere being parallellizable? Is that a stupid thing to say?
Being parallellizable seems like such a special property, I was wondering what the connections between different parallellizable manifolds are (other than, of course "there are not obstructions to this manifold being parallellizable")
 A: A large class of structures on manifolds are defined using "reduction of the structure group" applied to the tangent bundle: these include, but are not limited to, orientations, spin structures, almost complex structures, almost symplectic structures, etc.
A parallelizable manifold admits all of these structures. Is that the sort of thing you were looking for?
A: The first thing is purely pedagogical: A parallelizable manifold is the one which has trivial tangent bundle. When one is first introduced to (locally trivial) fiber bundles, trivial bundles are easy comparing to the general bundles; computations are sometimes easier for trivial bundles. (Later on you just get used to the concept.) 
Second, in the case of open manifolds $M$ (ones where each component is noncompact), there  is a beautiful theorem, due to Hirsch, which states that every open parallelizable $n$-dimensional manifold admits an local diffeomorphism $M\to R^n$. 
See 
Theorem 4.7 in
M. Hirsch, On imbedding differentiable manifolds in euclidean space. 
Ann. of Math. (2) 73 (1961) 566–571. 
and, with more details:
Corollary 8.2 in 
A. Phillips, Submersions of open manifolds. 
Topology 6 (1967) 171–206. 
Thus, you can take any structure you have on $R^n$, e.g. complex structure, flat Riemannian metric, etc., and pull back it to $M$.  
The most spectacular application of the Hirsch's immersion theorem is the "torus trick" of Kirby and Siebenmann. 
A: Parallelizable manifold is equivalent to the fact that the tangent bundle is trivial, so all the characteristic classes are zero, so is the euler number,... There is also a vector field which does not vanish at any point, a fact which is not true for every manifold.
The only parallelizable spheres are $S^1, S^3$ and $S^7$.
