Lie derivative (for vector fields)
LieDerivativeForVectorFields
2004-02-16 15:23:34
2006-09-16 22:45:05
Definition
Lie derivative
Lie derivative
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
\newtheorem{thm}{Theorem}
% define commands here
\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
\newcommand{\sZ}[0]{\mathbb{Z}}
\newcommand*{\norm}[1]{\lVert #1 \rVert}
\newcommand*{\abs}[1]{| #1 |}
\newcommand{\cbra}[1]{\left( #1 \right)}
\newcommand{\qbra}[1]{\left[ #1 \right]}
\newcommand{\gbra}[1]{\left\{ #1 \right\}}
\newcommand{\abra}[1]{\left\langle #1 \right\rangle}
\newcommand{\TTT}{\mathcal{T}}
\newcommand{\UUU}{\mathcal{U}}
\newcommand{\VVV}{\mathcal{V}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\LLL}{\mathcal{L}}
Let $M$ be a smooth manifold, and $X,Y\in\TTT(M)$ smooth vector fields
on $M$. Let $\Theta:\UUU\rightarrow M$ be the flow of $X$, where
$\UUU\subseteq \R\times M$ is an open neighborhood of
$\gbra{0}\times M$. We make use of the following notation:
$$\UUU^p=\gbra{t\in\R\,|\,(t,p)\in\UUU},\ \ \forall p\in M,$$
$$\UUU_t=\gbra{p\in M\,|\,(t,p)\in\UUU},\ \ \forall t\in\R,$$ and we introduce the
auxiliary maps $\theta_t:\UUU_t\rightarrow M$ and
$\theta^p:\UUU^p\rightarrow M$ defined as
$$\Theta(t,p)=\theta_t(p)=\theta^p(t),\ \ \forall (t,p)\in\UUU.$$
The \emph{Lie derivative} of $Y$ along $X$ is the vector field
$\LLL_XY\in\TTT(M)$ defined by
$$(\LLL_XY)_p=\left. \frac{d}{dt} \cbra{ d(\theta_{-t})_{\theta_t(p)} (Y_{\theta_t(p)}) } \right|_{t=0}
=\lim_{t\rightarrow0}\frac{d(\theta_{-t})_{\theta_t(p)}
(Y_{\theta_t(p)}) - Y_p}{t},\ \ \forall p\in M,$$
where $d(\theta_{-t})_{\theta_t(p)}\in\mathrm{Hom}(T_{\theta_{t}(p)}M,T_pM)$ 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.
\begin{thm}
$\LLL_XY=[X,Y]$, where $[X,Y]=XY-YX$ is the Lie bracket of $X$ and
$Y$.
\end{thm}