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[parent] Lie derivative (for vector fields) (Definition)

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:

$\displaystyle \mathcal{U}^p=\left\{ t\in\mathbb{R}\,\vert\,(t,p)\in\mathcal{U} \right\},\ \ \forall p\in M,$
$\displaystyle \mathcal{U}_t=\left\{ p\in M\,\vert\,(t,p)\in\mathcal{U} \right\},\ \ \forall t\in\mathbb{R},$
and we introduce the auxiliary maps $ \theta_t:\mathcal{U}_t\rightarrow M$ and $ \theta^p:\mathcal{U}^p\rightarrow M$ defined as
$\displaystyle \Theta(t,p)=\theta_t(p)=\theta^p(t),\ \ \forall (t,p)\in\mathcal{U}.$

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

$\displaystyle (\mathcal{L}_XY)_p=\left. \frac{d}{dt} \left( d(\theta_{-t})_{\th... ...ac{d(\theta_{-t})_{\theta_t(p)} (Y_{\theta_t(p)}) - Y_p}{t},\ \ \forall p\in M,$
where $ d(\theta_{-t})_{\theta_t(p)}\in\mathrm{Hom}(T_{\theta_{t}(p)}M,T_pM)$ if the push-forward of $ \theta_{-t}$, i.e.
$\displaystyle d(\theta_{-t})_{\theta_t(p)}(v)(f)=v(f\circ\theta_{-t}),\ \ \ \forall v\in T_{\theta_{-t}(p)}M,\ f\in C^\infty(p).$

The following result is not immediate at all.

Theorem 1   $ \mathcal{L}_XY=[X,Y]$, where $ [X,Y]=XY-YX$ is the Lie bracket of $ X$ and $ Y$.



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Cross-references: Lie bracket, maps, neighborhood, open, flow, vector fields, smooth, smooth manifold
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This is version 6 of Lie derivative (for vector fields), born on 2004-02-16, modified 2006-09-16.
Object id is 5590, canonical name is LieDerivativeForVectorFields.
Accessed 4132 times total.

Classification:
AMS MSC53-00 (Differential geometry :: General reference works )

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