Differentiable Manifolds/Lie algebras and the vector field Lie bracket

The vector field Lie bracket
Theorem 6.4: If $$\mathbf V, \mathbf W$$ are vector fields of class $$\mathcal C^n$$ on $$M$$, then $$[\mathbf V, \mathbf W]$$ is a vector field of class $$\mathcal C^n$$ on $$M$$ (i. e. $$[\cdot, \cdot]$$ really maps to $$\mathfrak X(M)$$)

Proof:

1. We show that for each $$p \in M$$, $$[\mathbf V, \mathbf W](p) \in T(p) M$$. Let $$\varphi, \vartheta \in \mathcal C^\infty(M)$$ and $$c \in \mathbb R$$.

1.1 We prove linearity:
 * $$\begin{align}

{[\mathbf V, \mathbf W]}(p) (\varphi + c \vartheta) & = \mathbf V(p)(\mathbf W (\varphi + c \vartheta)) - \mathbf W(p)(\mathbf V (\varphi + c \vartheta)) \\ & = \mathbf V(p)(\mathbf W \varphi + c \mathbf W \vartheta) - \mathbf W(p)(\mathbf V \varphi + c \mathbf V \vartheta) \\ & = \mathbf V(p)(\mathbf W \varphi) - \mathbf W(p)(\mathbf V \varphi) + c (\mathbf V(p)(\mathbf W \vartheta) - \mathbf W(p)(\mathbf V \vartheta)) \\ & = [\mathbf V, \mathbf W](p)(\varphi) + c [\mathbf V, \mathbf W](p)(\vartheta) \end{align}$$

1.2 We prove the product rule:
 * $$\begin{align}

{[\mathbf V, \mathbf W]}(p) (\varphi \vartheta) & = \mathbf V(p) (\mathbf W (\varphi \vartheta)) - \mathbf W(p) (\mathbf V (\varphi \vartheta)) \\ & = \mathbf V(p) (\varphi \mathbf W \vartheta + \vartheta \mathbf W \varphi) - \mathbf W(p) (\varphi \mathbf V \vartheta + \vartheta \mathbf V \varphi) \\ & = \mathbf V(p) (\varphi \mathbf W \vartheta) + \mathbf V(p) (\vartheta \mathbf W \varphi) - \mathbf W(p) (\varphi \mathbf V \vartheta) - \mathbf W(p) (\vartheta \mathbf V \varphi) \\ & = \varphi(p) \mathbf V(p) (\mathbf W \vartheta) + \overbrace{\mathbf (Y \vartheta)(p)}^{= Y(p)(\vartheta)} \mathbf V(p)(\varphi) + \vartheta(p) \mathbf V(p) (\mathbf W \varphi) + \overbrace{\mathbf (Y \varphi)(p)}^{= Y(p)(\varphi)} \mathbf V(p)(\vartheta) \\ & ~ - \varphi(p) \mathbf W(p) (\mathbf V \vartheta) - \overbrace{\mathbf (X \vartheta)(p)}^{= X(p)(\vartheta)} \mathbf W(p)(\varphi) - \vartheta(p) \mathbf W(p) (\mathbf V \varphi) - \overbrace{\mathbf (X \varphi)(p)}^{= X(p)(\varphi)} \mathbf W(p)(\vartheta) \\ & = \varphi(p) \mathbf V(p) (\mathbf W \vartheta) - \varphi(p) \mathbf W(p) (\mathbf V \vartheta) + \vartheta(p) \mathbf V(p) (\mathbf W \varphi)- \vartheta(p) \mathbf W(p) (\mathbf V \varphi) \\ & = \varphi(p) [\mathbf V, \mathbf W](p) (\vartheta) + \vartheta(p) [\mathbf V, \mathbf W](p) (\varphi) \end{align}$$

2. We show that $$[\mathbf V, \mathbf W]$$ is differentiable of class $$\mathcal C^n$$.

Let $$\varphi \in \mathcal C^n(M)$$ be arbitrary. As $$\mathbf V, \mathbf W$$ are vector fields of class $$\mathcal C^n$$, $$\mathbf V \varphi$$ and $$\mathbf W \varphi$$ are contained in $$\mathcal C^n(M)$$. But since $$\mathbf V, \mathbf W$$ are vector fields of class $$\mathcal C^n$$, $$\mathbf V (\mathbf W \varphi)$$ and $$\mathbf W (\mathbf V \varphi)$$ are contained in $$\mathcal C^n(M)$$. But the sum of two differentiable functions is again differentiable (this is what theorem 2.? says), and thus $$[\mathbf V, \mathbf W] \varphi$$ is in $$\mathcal C^n(M)$$, and since $$\varphi$$ was arbitrary, $$[\mathbf V, \mathbf W]$$ is differentiable of class $$\mathcal C^n$$.

Theorem 6.5:

If $$M$$ is a manifold, and $$[ \cdot, \cdot ]$$ is the vector field Lie bracket, then $$\mathfrak X(M)$$ and $$[ \cdot, \cdot ]$$ form a Lie algebra together.

Proof:

1. First we note that $$\mathfrak X(M)$$ as defined in definition 5.? is a vector space (this was covered by exercise 5.?).

2. Second, we prove that for the vector Lie bracket, the three calculation rules of definition 6.1 are satisfied. Let $$\mathbf V, \mathbf W, \mathbf U \in \mathfrak X(M)$$ and $$c \in \mathbb R$$.

2.1 We prove bilinearity. For all $$p \in M$$ and $$\varphi \in \mathcal C^n(M)$$, we have
 * $$\begin{align}

{[\mathbf V, \mathbf W + c \mathbf U]}(p) (\varphi) & = \mathbf V(p) ((\mathbf W + c \mathbf U) \varphi) - (\mathbf W + c \mathbf U)(p) (\mathbf V \varphi) \\ & = \mathbf V(p) (\mathbf W \varphi + c \mathbf U \varphi) - \mathbf W(p) (\mathbf V \varphi) - c \mathbf U(p) (\mathbf V \varphi) \\ & = \mathbf V(p) (\mathbf W \varphi) - \mathbf W(p) (\mathbf V \varphi) + c \mathbf V(p) (\mathbf U \varphi) - c \mathbf U(p) (\mathbf V \varphi) \\ & = [\mathbf V, \mathbf W](p) (\varphi) + c [\mathbf V, \mathbf U](p) (\varphi) \end{align}$$ and hence, since $$p \in M$$ and $$\varphi \in \mathcal C^n(M)$$ were arbitrary,
 * $$[\mathbf V, \mathbf W + c \mathbf U] = [\mathbf V, \mathbf W] + c [\mathbf V, \mathbf U]$$

Analogously (see exercise 1), it can be proven that
 * $$[\mathbf V + c \mathbf W, \mathbf U] = [\mathbf V, \mathbf U] + c [\mathbf W, \mathbf U]$$

2.2 We prove skew-symmetry. We have for all $$p \in M$$ and $$\varphi \in \mathcal C^n(M)$$:
 * $$[\mathbf V, \mathbf W](p)(\varphi) = \mathbf V(p) (\mathbf W \varphi) - \mathbf W(p) (\mathbf V \varphi) = - (\mathbf W(p) (\mathbf V \varphi) - \mathbf V(p) (\mathbf W \varphi)) = - [\mathbf W, \mathbf V](p)(\varphi)$$

2.3 We prove Jacobi's identity. We have for all $$p \in M$$ and $$\varphi \in \mathcal C^n(M)$$:
 * $$\begin{align}

{[\mathbf V, [\mathbf W, \mathbf U]]}(p)(\varphi) + [\mathbf U, [\mathbf V, \mathbf W]](p)(\varphi) + [\mathbf W, [\mathbf U, \mathbf V]](p)(\varphi) & = \mathbf V(p)([\mathbf W, \mathbf U] \varphi) - [\mathbf W, \mathbf U](p) (\mathbf V \varphi) \\ & ~ + \mathbf U(p)([\mathbf V, \mathbf W] \varphi) - [\mathbf V, \mathbf W](p) (\mathbf U \varphi) \\ & ~ + \mathbf W(p)([\mathbf U, \mathbf V] \varphi) - [\mathbf U, \mathbf V](p) (\mathbf W \varphi) \\ & = \mathbf V(p)(\mathbf W(\mathbf U \varphi) - \mathbf U(\mathbf W \varphi)) - \mathbf W(p) (\mathbf U (\mathbf V \varphi)) + \mathbf U(p)(\mathbf W(\mathbf V\varphi)) \\ & ~ + \mathbf U(p)(\mathbf V(\mathbf W \varphi) - \mathbf W(\mathbf V \varphi)) - \mathbf V(p) (\mathbf W (\mathbf U \varphi)) + \mathbf W(p)(\mathbf V(\mathbf U\varphi)) \\ & ~ + \mathbf W(p)(\mathbf U(\mathbf V \varphi) - \mathbf V(\mathbf U \varphi)) - \mathbf U(p) (\mathbf V (\mathbf W \varphi)) + \mathbf V(p)(\mathbf U(\mathbf W\varphi)) \\ & = 0 \end{align}$$ , where the last equality follows from the linearity of $$\mathbf V(p), \mathbf W(p)$$ and $$\mathbf U(p)$$.