# Lie derivative (for vector fields)

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

 $\mathcal{U}^{p}=\left\{t\in\mathbb{R}\,|\,(t,p)\in\mathcal{U}\right\},\ \ % \forall p\in M,$
 $\mathcal{U}_{t}=\left\{p\in M\,|\,(t,p)\in\mathcal{U}\right\},\ \ \forall t\in% \mathbb{R},$

and we introduce the auxiliary maps $\theta_{t}:\mathcal{U}_{t}\rightarrow M$ and $\theta^{p}:\mathcal{U}^{p}\rightarrow M$ defined as

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

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

 $(\mathcal{L}_{X}Y)_{p}=\left.\frac{d}{dt}\left(d(\theta_{-t})_{\theta_{t}(p)}(% Y_{\theta_{t}(p)})\right)\right|_{t=0}=\lim_{t\rightarrow 0}\frac{d(\theta_{-t% })_{\theta_{t}(p)}(Y_{\theta_{t}(p)})-Y_{p}}{t},\ \ \forall p\in M,$

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.

 $d(\theta_{-t})_{\theta_{t}(p)}(v)(f)=v(f\circ\theta_{-t}),\ \ \ \forall v\in T% _{\theta_{-t}(p)}M,\ f\in C^{\infty}(p).$

The following result is not immediate at all.

###### Theorem 1

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

Title Lie derivative (for vector fields) LieDerivativeforVectorFields 2013-03-22 14:09:59 2013-03-22 14:09:59 matte (1858) matte (1858) 9 matte (1858) Definition msc 53-00 Lie derivative