Machine-translated from Chinese.
Introduction to smooth manifolds by John M. Lee
定义
取关于$V$的流$\theta,$ 对$M$上的切向量场$X,$ 定义其关于$V$的李导数为:
$$
(L_VX)_p=\left.\frac{d {} }{d {}t}\right|_{t=0}d(\theta_{-t})_{\theta_t(p)}X_{\theta_t(p)}=\lim_{t\rightarrow 0}\frac{d(\theta_{-t})_{\theta_t(p)}X_{\theta_t(p)}-X_p}{t}.
$$
对$M$上的光滑共变张量场$A,$ 定义其关于$V$的李导数为:
$$
(L_VA)_p=\left.\frac{d {} }{d {}t}\right|_{t=0}(\theta_t^\ast A)_p=\lim_{t\rightarrow 0}\frac{(\theta_t)^\ast _p(A_{\theta_t(p)})-A_p}{t}.
$$
注意这和协变导数的差别是: 协变导数是平移地拉回求导, 而李导数是直接拉回.
李导数可定义在一般的张量场上, 得到的也是同样类型的张量场.
性质
定理 1. $L_VW=[V,W].$
需要特别注意的一点是是$L_VW$依赖于$V(p)$附近的取值, 而$D_VW$不是.
这是因为李导数在$V$位置不是$C^\infty$线性的.
关于李导数直接的性质有它关于张量积, 缩并满足Leibniz法则.
一些常用的好的性质是它在微分形式上的作用:
$$
L_V(df)=d(L_Vf)=d(Vf),\quad L_V(d\omega)=d(L_V\omega).
$$
$$
L_V(\omega\wedge\eta)=L_V\omega\wedge\eta+\omega\wedge L_V\eta.
$$
定理 2 (Cartan’s Magic Formula). $L_V\omega=i_V(d\omega)+d(i_V\omega)$
归纳法. 零维显然. 记$\omega=fdx^1\wedge\cdots\wedge dx^n,$
$\eta=fdx^2\wedge\cdots\wedge dx^n,$ 故$\omega=dx^1\wedge \eta.$
$$
\begin{aligned}
L_V\omega&=L_Vdx^1\wedge \eta+dx^1\wedge L_V\eta\\
&=d(Vx^1)\wedge \eta+dx^1\wedge( i_V(d\eta)+d(i_V\eta))
\end{aligned}
$$
由于$d$和内乘都是反导子, 我们有:
$$
i_V(d\omega)=-i_Vdx^1\wedge d\eta=-Vx^1\wedge d\eta +dx^1\wedge i_V(d\eta)
$$
$$
d(i_V\omega)=d(Vx^1\wedge\eta-dx^1\wedge i_V\eta)=d(Vx^1)\wedge \eta +Vx^1 \wedge d\eta+dx^1\wedge d(i_V\eta)
$$
由此即得到结论.
文章最后更新于 2022-10-07 13:21:51
Introduction to smooth manifolds by John M. Lee
definition
Take the flow $\theta,$ about $V$ and the tangent vector field $X,$ on $M$ and define its Lie derivative about $V$ as:
$$
(L_VX)_p=\left.\frac{d {} }{d {}t}\right|_{t=0}d(\theta_{-t})_{\theta_t(p)}X_{\theta_t(p)}=\lim_{t\rightarrow 0}\frac{d(\theta_{-t})_{\theta_t(p)}X_{\theta_t(p)}-X_p}{t}.
$$
For the smooth covariant tensor field $A,$ on $M$, define its Lie derivative with respect to $V$ as:
$$
(L_VA)_p=\left.\frac{d {} }{d {}t}\right|_{t=0}(\theta_t^\ast A)_p=\lim_{t\rightarrow 0}\frac{(\theta_t)^\ast _p(A_{\theta_t(p)})-A_p}{t}.
$$
Note that the difference between this and the covariant derivative is that the covariant derivative is pulled back in translation, while the Lie derivative is pulled back directly.
The Lie derivative can be defined on a general tensor field, and the same type of tensor field is obtained.
nature
Theorem 1. $L_VW=[V,W].$
One thing that needs special attention is that $L_VW$ depends on the values near $V(p)$, but $D_VW$ does not.
This is because the Lie derivative is not linear at $V$ position $C^\infty$.
The direct properties of Lie derivatives are that they are about tensor products, and they shrink and satisfy Leibniz's rule.
Some commonly used nice properties are its effects on differential forms:
$$
L_V(df)=d(L_Vf)=d(Vf),\quad L_V(d\omega)=d(L_V\omega).
$$
$$
L_V(\omega\wedge\eta)=L_V\omega\wedge\eta+\omega\wedge L_V\eta.
$$
Theorem 2 (Cartan’s Magic Formula). $L_V\omega=i_V(d\omega)+d(i_V\omega)$
Induction method. Zero-dimensional obvious. Note $\omega=fdx^1\wedge\cdots\wedge dx^n,$
$\eta=fdx^2\wedge\cdots\wedge dx^n,$ Therefore $\omega=dx^1\wedge \eta.$
$$
\begin{aligned}
L_V\omega&=L_Vdx^1\wedge \eta+dx^1\wedge L_V\eta\\
&=d(Vx^1)\wedge \eta+dx^1\wedge( i_V(d\eta)+d(i_V\eta))
\end{aligned}
$$
Since $d$ and internal multiplication are both antiderivatives, we have:
$$
i_V(d\omega)=-i_Vdx^1\wedge d\eta=-Vx^1\wedge d\eta +dx^1\wedge i_V(d\eta)
$$
$$
d(i_V\omega)=d(Vx^1\wedge\eta-dx^1\wedge i_V\eta)=d(Vx^1)\wedge \eta +Vx^1 \wedge d\eta+dx^1\wedge d(i_V\eta)
$$
From this the conclusion is drawn.
The article was last updated on 2022-10-07 13:21:51
