Suppose on an $N$-dimensional manofold $ \mathcal{M} $, there exists $N$ continuous vector fields, which are linearly independent everywhere on $\mathcal{M} $.
These $N$ vector fields are not necessarily commutative.
The question is, does their existence mean that there do exist $N$ linearly independent and commutative vector fields? Namely, can we always rectify the original vector fields?
It is not true in general, let $M$ be a compact semi-simple Lie group $n$-dimensional Lie, it is parallelizable, but there does not exists $n$-vertor fields $X_1,..,X_n$ of $M$ which commutes and does not vanish. The existence of such vector implies that there exists a covering $R^n\rightarrow M$, and this not true since for example the covering space of $SO(m)$ is a compact Lie group.