On a differentiable manifold, let $x^i$ be the coordinates of points on the manifold in a local parametrization. Then it is straightforward to show that the Lie bracket between vector fields defined by the coordinate curve vanishes, i.e., $\left[\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right] = 0$. I, however, find this very strange. It seems that given any linearly independent vector fields, $X, Y,\ldots$, one can always choose a local parametrization such that the vector fields $X, Y,\ldots$ are the associated basis of the parametrization. But that would mean that any two vector fields would have zero Lie brackets, which is clearly wrong.
So where is wrong with this reasoning? (It seems it has to be that not all linearly independent vector fields can be made into associated basis of a local parametrization. But why not? What is the obstruction?)
Thanks!
Best Answer
In fact, if $X_1,\ldots,X_n$ are linearly independant vector fields, there exist coordinates $x^i$ such that $X_i = \partial/\partial x^i$ if and only if $[X_i,X_j]=0$. A nice generalisation is Frobenius's theorem : there exists a submanifold tangent to your linearly independant vector fields if and only if their bracket always vanish.