Let $X$ and $Y$ two vector field on the manifold $M$ (dim($M$)= $m$). Show that the Lie bracket $[X, Y] := XY – YX $ is a vector field on $M$.
I tried to compute using local coordinates, so for $X = \sum_{i=1}^{m} X_i \frac{\partial}{\partial x_i}$ and $Y = \sum_{i=1}^{m} Y_i \frac{\partial}{\partial x_i}$ I obtain after computation that
\begin{align}
[X, Y] = \sum_{i=1}^{m} \left( \sum_{j=1}^{m} X_j \frac{\partial Y_i}{\partial x_j} – Y_j \frac{\partial X_i}{\partial x_j} \right) \frac{\partial}{\partial x_i}
\end{align}
But how can I conclude now that the Lie bracket $[X,Y]$ is a vector field on $M$?
Thanks in advance!
Best Answer
Assuming everything is smooth:
$\textbf{Lemma}.$ Let $M$ be a smooth manifold. The map $D: C^{\infty}(M) \to C^{\infty}(M)$ is a derivation if and only if, $Df = Xf$ for some smooth vector field $X$.
In particular, this says that derivations of $C^{\infty}(M)$ can be identified with smooth vector fields. So, it suffices to show that the Lie bracket is a derivation of $C^{\infty}(M)$.
$\textbf{Sketch.}$ $$ \begin{align*} [X,Y](fg) &= X(Y(fg) - Y(X(fg)) \\ &= X(fYg + gYf) - Y(fXg + gXf) \\ &= \ldots \end{align*} $$