# 李导数

${\displaystyle [A,B]{\overset {def}{=}}{\mathcal {L}}_{A}B-{\mathcal {L}}_{B}A}$

## 定义

${\displaystyle {\mathcal {L}}_{X}f(p)=df(p)\,[X(p)]}$

${\displaystyle df={\frac {\partial f}{\partial x^{a}}}dx^{a}}$ .

${\displaystyle {\frac {d\gamma }{dt}}(t)=X(\gamma (t))}$

${\displaystyle {\mathcal {L}}_{X}f(p)={\frac {d}{dt}}f(\gamma (t))\vert _{t=0}}$ .

${\displaystyle X=X^{a}{\frac {\partial }{\partial x^{a}}}}$

${\displaystyle [X,Y]=X^{a}{\frac {\partial Y^{b}}{\partial x^{a}}}{\frac {\partial }{\partial x^{b}}}-Y^{a}{\frac {\partial X^{b}}{\partial x^{a}}}{\frac {\partial }{\partial x^{b}}}}$

${\displaystyle {\mathcal {L}}_{X}Y=[X,Y]}$ .

${\displaystyle {\mathcal {L}}_{X}(f)=df(X)=X(f)}$

${\displaystyle [X,Y]f=X(Y(f))-Y(X(f))}$ .

${\displaystyle {\mathcal {L}}_{X}\omega =\left({\frac {\partial \omega _{b}}{\partial x^{a}}}X^{a}+{\frac {\partial X^{a}}{\partial x^{b}}}\omega _{a}\right)dx^{b}}$ .

## 性质

${\displaystyle {\mathcal {L}}_{X}:{\mathcal {F}}(M)\rightarrow {\mathcal {F}}(M)}$

${\displaystyle {\mathcal {L}}_{X}(fg)=({\mathcal {L}}_{X}f)g+f{\mathcal {L}}_{X}g}$ .

${\displaystyle {\mathcal {L}}_{X}(fY)=({\mathcal {L}}_{X}f)Y+f{\mathcal {L}}_{X}Y}$

${\displaystyle {\mathcal {L}}_{X}(f\otimes Y)=({\mathcal {L}}_{X}f)\otimes Y+f\otimes {\mathcal {L}}_{X}Y}$

${\displaystyle {\mathcal {L}}_{X}[Y,Z]=[{\mathcal {L}}_{X}Y,Z]+[Y,{\mathcal {L}}_{X}Z]}$

## 和外导数的关系、微分形式的李导数

M为一个流形，XM上一个向量场。令${\displaystyle \omega \in \Lambda ^{k+1}(M)}$ 为一k+1-形式。 X和ω的内积

${\displaystyle i_{X}\omega (X_{1},\ldots ,X_{k})=\omega (X,X_{1},\ldots ,X_{k})}$

${\displaystyle i_{X}:\Lambda ^{k+1}(M)\rightarrow \Lambda ^{k}(M)}$

${\displaystyle i_{X}(\omega \wedge \eta )=(i_{X}\omega )\wedge \eta +(-1)^{k}\omega \wedge (i_{X}\eta )}$

${\displaystyle i_{fX}\omega =fi_{X}\omega }$

${\displaystyle {\mathcal {L}}_{X}f=i_{X}df}$

${\displaystyle {\mathcal {L}}_{X}\omega =i_{X}d\omega +d(i_{X}\omega )}$ .

${\displaystyle d\omega (X,Y)=X(\omega (Y))-Y(\omega (X))-\omega ([X,Y]).}$

${\displaystyle {\mathcal {L}}_{fX}\omega =f{\mathcal {L}}_{X}\omega +df\wedge i_{X}\omega }$

## 张量场的李导数

${\displaystyle T(f_{1}\alpha ,f_{2}\beta ,\ldots ,f_{p+1}X,f_{p+2}Y,\ldots )=f_{1}f_{2}\cdots f_{p+1}f_{p+2}\cdots f_{p+q}T(\alpha ,\beta ,\ldots ,X,Y,\ldots )}$ ),

${\displaystyle ({\mathcal {L}}_{A}T)(\alpha ,\beta ,\ldots ,X,Y,\ldots )\equiv \nabla _{A}T(\alpha ,\beta ,\ldots ,X,Y,\ldots )-\nabla _{T(\cdot ,\beta ,\ldots ,X,Y,\ldots )}\alpha (A)-\ldots +T(\alpha ,\beta ,\ldots ,\nabla _{X}A,Y,\ldots )+\ldots }$

${\displaystyle {\mathcal {L}}_{U}T={\frac {d}{dt}}\left(\psi _{t}^{*}T\right)\vert _{\psi (t)=p}}$ .

## 参考

• Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-4267-2 See section 1.6.
• Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 2.2.
• David Bleecker, Gauge Theory and Variational Principles, (1981), Addison-Wesley Publishing, ISBN 0-201-10096-7. See Chapter 0.