微分流形上微分学—流形上的微分运算—Lie导数 谢锡麟 故有 LIZ=(LoL LZ az i auP Oazp(az)(Org(z)u- Oxg(a)u )+Zp auP(ac)ug au au duq dup d rq LLu, LuZ 2.首先考虑 Lu=Itu,可(Zg2) 0Z1 P (a)u,vPp ar2 (a)Zp az: auP aup a-uP (aa8-bm10)+p(aaa-a(a Ouq.、OvP aug aup +Zp(o a xi axe axk axq 所以有 LLu, Lu]z=Lu, Z 因此 ILLu, Lu,Luz=[Lu, Lu] LLu, u, Lu z=Lu, u, uZ 故有 (ILLu, Lu, Lu+[Lu, Lu], Lu+[Lw, Lu,Lv)Z (L+L+Lm)Z=[t,,+,,]+[,小,可,Z=0. 因此有 [Lu, Lu, Lw]+[Lu, Lu], Lu]+[[Lw, Lu], Lv]=0 最后,考虑一般情形更=j91⑧9,即根据定义 L,Lu](j91②9)=(LnL- Ly o lu)(重,g9;⑧91) 计算 Ln0L(9;8g)=Ln[(Lnv)91892+重,L(918g) (LnoL)9:893+(L重)Ln(9;893)+(L)Lu(91②93) 中. Luo ly(g1⑧9)微分流形上微分学 微分流形上微分学 —— 流形上的微分运算— Lie 导数 谢锡麟 故有 [Lu, Lv]Z = (Lu ◦ Lv − Lv ◦ Lu)Z = [ ∂Zi ∂xp (x) ( ∂vp ∂xq (x)u q − ∂up ∂xq (x)v q ) + Zp ( ∂ 2 v p ∂xq∂xi (x)u q − ∂ 2u p ∂xq∂xi (x)v q ) + Zp ( ∂uq ∂xi (x) ∂vp ∂xq (x) − ∂vq ∂xi (x) ∂up ∂xq (x) )] g i = −[Lv, Lu]Z. 2. 首先考虑 L[u,v]Z = L[u,v] (Zig i ) = ( ∂Zi ∂xp (x)[u, v] p + ∂[u, v] p ∂xi (x)Zp ) g i = [ ∂Zi ∂xp (x) ( ∂vp ∂xq (x)u q − ∂up ∂xq (x)v q ) + Zp ( ∂ 2v p ∂xq∂xi (x)u q − ∂ 2u p ∂xq∂xi (x)v q ) + Zp ( ∂uq ∂xi (x) ∂vp ∂xq (x) − ∂vq ∂xi (x) ∂up ∂xq (x) )] g i , 所以有 [Lu, Lv]Z = L[u,v]Z. 因此 [[Lu, Lv], Lw]Z = [Lu, Lv] ◦ LwZ − Lw ◦ [Lu, Lv]Z = L[u,v] (LwZ) − Lw ◦ L[u,v]Z = [L[u,v] , Lw]Z = L[[u,v],w]Z. 故有 ([[Lu, Lv], Lw] + [[Lv, Lw], Lu] + [[Lw, Lu], Lv]) Z = ( L[[u,v],w] + L[[v,w],u] + L[[w,u],v] ) Z = [[[u, v], w] + [[v, w],u] + [[w,u], v], Z] = 0. 因此有 [[Lu, Lv], Lw] + [[Lv, Lw], Lu] + [[Lw, Lu], Lv] = 0. 最后, 考虑一般情形 Φ = Φ i ·jgi ⊗ g j , 即根据定义 [Lu, Lv](Φ i ·jgi ⊗ g j ) = (Lu ◦ Lv − Lv ◦ Lu)(Φ i ·jgi ⊗ g j ). 计算 Lu ◦ Lv(Φ i ·jgi ⊗ g j ) = Lu [ (LvΦ i ·j )gi ⊗ g j + Φ i ·jLv(gi ⊗ g j ) ] = (Lu ◦ LvΦ i ·j )gi ⊗ g j + (LvΦ i ·j )Lu(gi ⊗ g j ) + (LuΦ i ·j )Lv(gi ⊗ g j ) + Φ i ·jLu ◦ Lv(gi ⊗ g j ), 18