TITLE: When does a SES of vector bundles split?
QUESTION [14 upvotes]: Given a short exact sequence of smooth vector bundles, 
$$0\to A \to B \to C \to 0$$ 
on a manifold $M$, it is an easy exercise that sequence splits. One approach is to pick a Riemannian metric on $B$ and show that $C$ is isomorphic to the orthogonal complement of $A$.  This proof extends to complex line bundles by choosing a Hermitian metric. 
If we leave the category of smooth bundles, a lack of bump functions means we can no longer assume a metric exists, although if we consider bundles equipped with a metric, the same proof should work. 
Question 1: Is there a proof that does not make use of a metric?
Question 2: In what generality does it hold that short exact sequences of vector bundles split? I know that vector bundles correspond to projective modules, which says that we have splittings over affine schemes, but what about more generally?

REPLY [4 votes]: I believe you can just use the Serre-Swan theorem, which gives an equivalence of categories between finite rank vector bundles over a smooth manifold and finitely generated projective modules over the ring of smooth functions. Since any short exact sequence of projective modules splits, you have the desired conclusion.
As to the generality of this method, there is a paper by Morye which discusses various ringed spaces where Serre-Swan type theorems hold.