# 李导数

$[A,B]:= \mathcal{L}_A B = - \mathcal{L}_B A$

## 定义

$\mathcal{L}_Xf(p)=df(p)\, [X(p)]$

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

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

$\mathcal{L}_Xf(p)=\frac{d}{dt} f(\gamma(t)) \vert_{t=0}$.

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

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

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

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

$[X,Y]f = X(Y( f )) - Y(X( f ))$.

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

## 性质

$\mathcal{L}_X : \mathcal{F}(M) \rightarrow \mathcal{F}(M)$

$\mathcal{L}_X(fg)=(\mathcal{L}_Xf) g + f\mathcal{L}_Xg$.

$\mathcal{L}_X(fY)=(\mathcal{L}_Xf) Y + f\mathcal{L}_X Y$

$\mathcal{L}_X(f\otimes Y)= (\mathcal{L}_Xf) \otimes Y + f\otimes \mathcal{L}_X Y$

$\mathcal{L}_X [Y,Z] = [\mathcal{L}_X Y,Z] + [Y,\mathcal{L}_X Z]$

## 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$\omega \in \Lambda^{k+1}(M)$为一k+1-形式。 X和ω的内积

$i_X\omega (X_1,\ldots,X_k) = \omega (X,X_1,\ldots,X_k)$

$i_X:\Lambda^{k+1}(M) \rightarrow \Lambda^k(M)$

$i_X (\omega \wedge \eta) = (i_X \omega) \wedge \eta + (-1)^k \omega \wedge (i_X \eta)$

$i_{fX} \omega = fi_X\omega$

$\mathcal{L}_Xf = i_X df$

$\mathcal{L}_X\omega = i_Xd\omega + d(i_X \omega)$.

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

$\mathcal{L}_{fX}\omega = f\mathcal{L}_X\omega + df \wedge i_X \omega$

## 张量场的李导数

$T(f_1\alpha,f_2\beta,\ldots,f_{p+1}X,f_{p+2}Y,\ldots) = f_1 f_2 \cdots f_{p+1} f_{p+2} \cdots f_{p+q} T(\alpha,\beta,\ldots,X,Y,\ldots)$),

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

$\mathcal{L}_U T = \frac{d}{dt}\left(\psi^*_t T\right) \vert_{\psi(t)=p}$.

## 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.