微分流形上微分学一流形上的微分运算一Le导数 谢锡麟 即有 vlo: g gj (c) Lv(uo.)-「on2 9 Laze (a)V+ o(aJulg' 性质1.5(Lie导数的运算性质) al 证明不失一般性,考虑更=更91893,故有 更=Lv(./91g3)(a) O (e)V 故有 yφ(u, p.(x)3+ aV. axl =V更(u,)(x)-V(x)+ act a)2 v(更(u,)-(Lv,2v)-更( 性质1.6. 1.对V重,亚∈P(T∑),Va,B∈R,有 (a更+)=aLv更+BLv重; 对Ⅵf∈6(∑),v更∈P(T∑)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 即有 LV f = V (f), LV gi = LV (δ j i gj ) = [ ∂δj i ∂xl (x)V l − ∂V j ∂xl (x)δ l i ] gj = − ∂V j ∂xi (x)gj , LV g i = LV (δ i jg j ) = [ ∂δi j ∂xl (x)V l + ∂V l ∂xj (x)δ i l ] g j = ∂V i ∂xj (x)g j , LV (u i gi ) = [ ∂ui ∂xl (x)V l − ∂V i ∂xl (x)u l ] gi = [V ,u], LV (uig i ) = [ ∂ui ∂xl (x)V l + ∂V l ∂xi (x)ul ] g i = [ V l ∂ui ∂xl (x) + ul ∂V l ∂xi (x) ] g i . 性质 1.5 (Lie 导数的运算性质). LV Φ(u, v) = V (Φ(u, v)) − Φ(LV u, v) − Φ(u, LV v). 证明 不失一般性, 考虑 Φ = Φ i ·jgi ⊗ g j , 故有 LV Φ = LV (Φ i ·jgi ⊗ g j )(x) = [ ∂Φi ·j ∂xl (x)V l − ∂V i ∂xl (x)Φ l ·j + ∂V l ∂xj (x)Φ i ·l ] (gi ⊗ g j )(x), 故有 LV Φ(u, v) = [ ∂Φi ·j ∂xl (x)V l − ∂V i ∂xl (x)Φ l ·j + ∂V l ∂xj (x)Φ i ·l ] uiv j = V l ∂ ∂xl (Φ i ·juiv j )(x) − V lΦ i ·j ∂ui ∂xl (x)v j − V lΦ i ·jui ∂vj ∂xl (x) − Φ i ·j ∂V l ∂xi (x)ulv j + Φ i ·j ∂V j ∂xl (x)uiv l = V l ∂ ∂xl Φ(u, v)(x) − Φ i ·j [ V l ∂ui ∂xl (x) + ul ∂V l ∂xi (x) ] v j − Φ i ·jui [ V l ∂vj ∂xl (x) − v l ∂V j ∂xl (x) ] = V (Φ(u, v)) − Φ(LV u, v) − Φ(u, LV v). 性质 1.6. 1. 对 ∀ Φ, Ψ ∈ T p (T Σ), ∀ α, β ∈ R, 有 LV (αΦ + βΨ) = αLV Φ + βLV Ψ; 2. 对 ∀ f ∈ C ∞(Σ), ∀ Φ ∈ T p (T Σ), 有 LV (fΦ) = (LV f)Φ + fLV Φ; 13