Lie derivative (for vector fields)

Let $M$ be a smooth manifold, and $X,Y\in 𝒯(M)$ smooth vector fieldson $M$. Let $\mathrm{\Theta }:𝒰\to M$ be the flow of $X$, where$𝒰\subseteq ℝ\times M$ is an open neighborhood of$\left\{0\right\}\times M$. We make use of the following notation:

$${𝒰}^p=\{t\in ℝ|(t,p)\in 𝒰\},\forall p\in M,$$

$${𝒰}_t=\{p\in M|(t,p)\in 𝒰\},\forall t\in ℝ,$$

and we introduce theauxiliary maps ${\theta }_t:{𝒰}_t\to M$ and${\theta }^p:{𝒰}^p\to M$ defined as

$$\mathrm{\Theta }(t,p)={\theta }_t(p)={\theta }^p(t),\forall (t,p)\in 𝒰.$$

The Lie derivative of $Y$ along $X$ is the vector field${ℒ}_{X}Y\in 𝒯(M)$ defined by

$${({ℒ}_{X}Y)}_p={\frac{d}{dt}\left(d{({\theta }_{-t})}_{{\theta }_t(p)}(Y_{{\theta }_t(p)})\right)|}_{t=0}=\underset{t\to 0}{lim}\frac{d{({\theta }_{-t})}_{{\theta }_t(p)}(Y_{{\theta }_t(p)})-Y_p}{t},\forall p\in M,$$

where $d{({\theta }_{-t})}_{{\theta }_t(p)}\in \mathrm{Hom}(T_{{\theta }_t(p)}M,T_{p}M)$ if the push-forward of ${\theta }_{-t}$, i.e.

$$d{({\theta }_{-t})}_{{\theta }_t(p)}(v)(f)=v(f\circ {\theta }_{-t}),\forall v\in T_{{\theta }_{-t}(p)}M,f\in C^{\mathrm{\infty }}(p).$$

The following result is not immediate at all.

Theorem 1

${ℒ}_{X}Y=[X,Y]$, where $\mathrm{[}X\mathrm{,}Y\mathrm{]}\mathrm{=}X\mathit{}Y\mathrm{-}Y\mathit{}X$ is the Lie bracket of $X$ and$Y$.