I have read that a short exact sequence of differentiable vector bundles is always split. I was interpreting this as the splitting being fiber wise, i.e that the fibers of the middle component of the short exact sequence are isomorphic as a $\mathbb{C}$-module to the direct sum of the fibers of the outside components.
However, I read that short exact sequences of holomorphic vector bundles are not necessarily split. This makes me believe that my fiber wise interpretation is wrong since the fibers are always vector spaces over $\mathbb{C}$ and thus being split would not depend on the vector bundle being holomorphic or not.
What is wrong with my original interpretation of what it means for a sequence of vector bundles to be split?
Best Answer
If you have an exact sequence $0\rightarrow E\rightarrow F\rightarrow G\rightarrow 0$ of vector bundles over the manifold $M$. Take a differentiable metric $b$ on $F$, and the orthogonal of $i(E)$ is isomorphic to $G$. You don't have necessarily holomorphic metric on complex vector bundles.
There is an old paper of Atiyah (I believe) which constructs a class which is the obstruction to the existence of such a splitting for complex bundles.