PlanetMath: Lie derivative (for vector fields)

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (230)
Orphanage
Unclass'd
Unproven (519)
Corrections (16)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

Lie derivative (for vector fields)

(Definition)

Let be a smooth manifold, and $X,Y\in\mathcal{T}(M)$ smooth vector fields on . Let $\Theta:\mathcal{U}\rightarrow M$ be the flow of , 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 along 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 is the Lie bracket of and .

Anyone with an account can edit this entry. Please help improve it!

"Lie derivative (for vector fields)" is owned by matte. [ full author list (3) ]

(view preamble)

Also defines:

Lie derivative

This object's parent.

Log in to rate this entry.
(view current ratings)

Cross-references: Lie bracket, maps, neighborhood, open, flow, vector fields, smooth, smooth manifold
There are 2 references to this entry.

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 MSC:

53-00 (Differential geometry :: General reference works )

Pending Errata and Addenda

None.[ View all 1 ]

Discussion

forum policy

No messages.

Interact