# Transition functions of trivial vector bundle

I have a question on trivial vector bundles. The question is as follows:

Can we characterize the transition functions of a trivial vector bundle in some way?

To be very concrete: suppose we have a vector bundle $E$, say of rank $3$, on an algebraic variety $X$ (or manifold if you prefer), and we look at trivial sub-bundles $0\to F\to E$ of rank $2$. What can we say about the transition functions of $F$ if we know those of $E$?

In this particular circumstance, we do have $3\times 3$ matrices at hand, and we want to produce $2\times 2$ matrices. The question is which ones are good.

At the beginning, I thought I could extend an argument of pure linear algebra: if we have an exact sequence of vector spaces $0\to V\to W\to W/V\to 0$, we can complete a basis $(e_1,e_2)$ of $V$ to a basis $(e_1,e_2,e_3)$ of $W$, so that $W\to W/V$ is represented by the row vector $e_3$ and $V\to W$ by the $3\times 2$ matrix containing $e_1,e_2$ written as columns. This didn't bring me anywhere, even because I can't figure how to use the triviality hypothesis.

A bundle is trivial if and only if the cocycle of its transition functions becomes cohomologous to the trivial cocycle $1$ after suitable refinement of the covering on which the transition functions are defined. I don't think one gets any additional information from knowing that the trivial bundle $F$ in question is a subbundle of a larger (possibly nontrivial) bundle $E$. Also, it is not true in general that a basis of sections for a subbundle can be extended globally to a basis for a larger bundle.

• Yes, I intended to use that linear algebra fact locally on $0\to F_x\to E_x\to E_x/F_x\to 0$. However, what is the trivial cocycle $1$? Do you mean a coboundary?
– Jack
Jun 10, 2013 at 15:09
• The trivial cocycle 1 is the cocycle whose values are all equal to 1. A cocycle it cohomologous to the trivial cocycle if and only if it is a coboundary. Jun 11, 2013 at 16:20
• Dear Andreas, thank you. I was hoping to understand in a more explicit way trivial sub-bundles of a given bundle.
– Jack
Jun 12, 2013 at 16:16

Vector bundle is trivial iff the transition functions have the form $$g_{ij}=g_{i}g_{j}^{-1}$$ where $$g_i \in {\rm{GL}}(n,\mathcal{O}_{U_i})$$.

Composite the trivilization with $$g_i^{-1}$$ we get another trivilization and the new transition functions are $$g_{ij}\equiv{1}$$.