微分流形上微分学—流形上的微分运算—Lie导数 谢锡麟 1.根据定义,有 LLu, LyZ=( 由于 LuZ=Luv, z=a, v, zI, 根据{u,Z]+{u,z,u+[z,,可]=0,则有 [Z,u, u=la,ul, Z=-Lu, LuZ 2.根据定义,有 ILLu, Lu, Luz=[Lu, Lyo Luz-[ Lu, LuZ LLu, Luz=Llu,ul uo Z, 故有 DZ (Lt+L[m,+Lma可)z=[u,],]+[,],u4+[v,u,v,Z1=0. 因此有 [Lu, Lu], Lw]+[Lu, Lu], Lu]+[Lw, Lu], Lv]=0 上述关于向量值函数Z=Z92的分析,亦可考虑如下 1.计算 Lu o L,Z=Luo Lu(219)=Lu( azp (a)u9+o(a)Zp9 (a)ug)+Lu di(2)z an((a))()n9+{a)2 a/dup Z axq ax2 (a)a(a)Zp g 02z auq axg arp Oug aup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 1. 根据定义, 有 [Lu, Lv]Z = (Lu ◦ Lv − Lv ◦ Lu)Z. 由于 Lu ◦ LvZ = Lu[v, Z] = [u, [v, Z]], 根据 [u, [v, Z]] + [v, [Z,u] + [Z, [u, v]] = 0, 则有 [Lu, Lv]Z = [u, [v, Z]] − [v, [u, Z]] = −[Z, [u, v]] = [[u, v], Z] = −[Lv, Lu]Z. 2. 根据定义, 有 [[Lu, Lv], Lw]Z = [Lu, Lv] ◦ LwZ − Lw ◦ [Lu, Lv]Z = L[u,v] (LwZ) − Lw ◦ L[u,v]Z = (L[u,v] ◦ Lw − 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. 上述关于向量值函数 Z = Zig i 的分析, 亦可考虑如下: 1. 计算 Lu ◦ LvZ = Lu ◦ Lv(Zig i ) = Lu ( ∂Zi ∂xp (x)v p g i + ∂vp ∂xi (x)Zpg i ) = Lu ( ∂Zi ∂xp (x)v p g i ) + Lu ( ∂vp ∂xi (x)Zpg i ) = [ ∂ ∂xq ( ∂Zi ∂xp (x)v p ) (x)u q + ∂uq ∂xi (x) ∂Zq ∂xp (x)v p ] g i + [ ∂ ∂xq ( ∂vp ∂xi (x)Zp ) (x)u q + ∂uq ∂xi (x) ∂vp ∂xq (x)Zp ] g i = [ ∂ 2Zi ∂xq∂xp (x)v pu q + ∂Zi ∂xp (x) ∂vp ∂xq (x)u q + ∂uq ∂xi (x) ∂Zq ∂xp (x)v p + ∂ 2 v p ∂xq∂xi (x)Zpu q + ∂vp ∂xi (x) ∂Zp ∂xq (x)u q + ∂uq ∂xi (x) ∂vp ∂xq (x)Zp ] g i , 17