I understand the definition of the Lie bracket and I know how to compute it in local coordinates.
But is there a way to “guess” what is the Lie bracket of two vector fields ? What is the geometric intuition ?
For instance, if we take U=x∂∂x+y∂∂y and V=−y∂∂x+x∂∂y, should it be obvious that [U,V]=0 ?
One way to get a geometric intuition for the Lie bracket is to note Φ∗([U,V])=[Φ∗(U),Φ∗(V)], i.e. the Lie bracket transforms canonically under the diffeomorphism Φ. Now if we have a straightening ΦU of the vector field U (such that ΦU∗(U) is constant in our coordinate system), then [U,V] is just the derivative of V along (the constant direction) U in that coordinate system.