When does a SES of vector bundles split?

Question

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?

Answer 1

Not totally sure what you're asking (particularly, are you asking about splitting in algebraic/holomorphic settings?), but the following comments may be relevant:

Because of the principle that "curvature decreases in holomorphic sub-bundles and increases in quotient bundles", it's "rare" for a short exact sequence of holomorphic vector bundles to split holomorphically. For example, the sequence $$ 0 \to \mathcal{O}(-1) \to \mathbf{C}^{n+1} \to T\mathbf{P}^{n}(-1) \to 0 $$ over the complex projective space $\mathbf{P}^{n}$ does not split holomorphically. In fact, the trivial bundle in the middle is split only by trivial sub-bundles.

Metrics don't help in the holomorphic category, because the components of a non-constant Hermitian metric are never holomorphic functions in local holomorphic coordinates.

Answer 2

I can't say I'm aware of a proof that doesn't use a metric, but (in relation to Question 2), this is probably because the proof by means of a Riemannian or Hermitian metric is actually very general. applying to all paracompact locally compact Hausdorff spaces (e.g., compact Hausdorff spaces, CW complexes).

If $X$ is just a locally compact Hausdorff space, then you can construct a continuous partition of unity subordinate to any finite cover by means of Urysohn's Lemma. In particular, if $X$ is paracompact (every open cover admits a locally finite refinement), e.g., if $X$ is compact or if $X$ is a CW complex, then you can construct a continuous partition of unity subordinate to any open cover at all. So, over a paracompact locally compact Hausdorff space, you can can always construct Riemannian or Hermitian metrics as needed and hence construct a splitting.

Now, even if you're interested in locally compact Hausdorff spaces, without requiring paracompactness, it's worth noting that in many contexts, e.g., topological $K$-theory, one only considers vector bundles trivial at infinity (i.e., $E \to X$ such that for some compact $K \subset X$, $E|_{X \setminus K}$ is trivial), in which case you can always pass to a finite trivialising subcover, which therefore admits a continuous partition of unity. Hence, a short exact sequence of vector bundles trivial at infinity, over a locally compact Hausdorff space, should split.