Is the Lie bracket of two vector fields well defined?

Question

I want to understand what exactly means to ask the question if the Lie bracket $[X,Y]$ of two vector fields $X,Y\in \mathcal{XM}$, where $\mathcal{M}$ is a differentiable manifold, is well defined.

• What does it mean conceptually that the Lie bracket is well defined?
• How could I prove that indeed it is well defined?
• Can I do it use it a specific basis, say, $e_x = \partial/\partial x$ and $e_y = \partial / \partial y$?

Answer 1

If $X$ and $Y$ are two vector fields, $[X,Y]=DY(X)-DX(Y)$. Here $DX$ is the differential of $X$.

You can see this https://en.wikipedia.org/wiki/Lie_bracket_of_vector_fields

Answer 2

The standard definition of the Lie bracket is: For a differentiable function $f$ on $M$ and two vector fields $X, Y$ on $M$, $$ [X,Y](f)= XY(f) - YX(f). $$ You need to check that this defines a derivation on the space of functions on $M$. Doing so is a very nice exercise in the definitions (the more interesting part is verifying the Leibnitz rule).