INTRODUCTION TO MANIFOLDS | III Algebra of vector fields. Lie derivative(s). 1. Notations. The space of all C1-smooth vector ¯elds on a manifold M is n denoted by X(M). If v 2 X(M) is a vector ¯eld, then v(x) 2 TxM ' R is its value at a point x 2 M. The flow of a vector ¯eld v is denoted by vt: 8t 2 R vt : M ! M is a smooth map (automorphism) of M taking a point x 2 M into the point vt(x) 2 M which is the t-endpoint of an integral trajectory for the ¯eld v, starting at the point x. | Problem 1. Prove that the flow maps for a ¯eld v on a compact manifold M form a one-parameter group: 8t; s 2 R vt+s = vt ± vs = vs ± vt; and all vt are di®eomorphisms of M. | Problem 2. What means the formula ¯ ¯ d ¯ s ¯ v = v ds s=0 and is it true? 2. Star conventions. The space of all C1-smooth functions is denoted by C1(M). If F : M ! M is a smooth map (not necessary a di®eomorphism), then there appears a contravariant map F ¤ : C1(M) ! C1(M);F ¤ : f 7! F ¤f; F ¤(x) = f(F (x)): If F : M ! N is a smooth map between two di®erent manifolds, then F ¤ : C1(N) ! C1(M): Note that the direction of the arrows is reversed! | Problem 3. Prove that C1(M) is a commutative associative algebra over R with respect to pointwise addition, subtraction and multiplication of functions. Prove that F ¤ is a homomorphism of this algebra (preserves all the operations). If F : M ! N, then F ¤ : C1(N) ! C1(M) is a homomorphism also. Typeset by AMS-TEX 1 2 ALGEBRA OF VECTOR FIELDS. LIE DERIVATIVE(S). Another star is associated with di®erentials: if F : M1 ! M2 is a diffeomor- phism, then @F F : X(M ) ! X(M ); v 7! F v; (F v)(x) = (x) ¢ v(x); ¤ 1 2 ¤ ¤ @x is a covariant (acts in the same direction) map which is: (1) additive: F¤(v + w) = F¤v + F¤w; 1 ¤ ¡1 (2) homogeneous: 8f 2 C (M) F¤(fv) = (F ) f ¢ F¤v. (explain this for- mula!), Why F¤ is in general not de¯ned, if F is just a smooth map and not a di®eomor- phism? 3. Vector ¯elds as di®erential operators. ~ De¯nition. If v 2 X(M), then the Lie derivative Lv is 1 1 1 ¡ t ¤ ¢ Lv : C (M) ! C (M);Lvf = lim (v ) f ¡ f : t!0 t In coordinates: f(a + tv + o(t)) ¡ f(a) Xn @f Lvf(a) = lim = (a)vj: t!0 t @x j=1 j Properties of the Lie derivative: 1 1 (1) Lv : C (M) ! C (M) is a linear operator: Lv(f + g) = Lvf + Lvg; Lv(¸f) = ¸Lvf; (2) the Leibnitz identity holds: Lv(fg) = Lvf ¢ g + f ¢ Lvg: (3) The Lie derivative linearly depends on v: 1 8f 2 C (M); v; w 2 X(M) Lfv = fLv;Lv+w = Lv + Lw: | Problem 4. Prove that the Lie derivative is local: for any function f 2 C1(M) and any vector ¯eld v the value Lvf(a) depends only on v(a), so that for any other ¯eld w such that w(a) = v(a), Lvf(a) = Lwf(a). Theorem. Any di®erential operator, that is, a map D : C1(M) ! C1(M) satis- fying D(f + g) = Df + Dg; D(¸f) = ¸Df; D(fg) = f Dg + Df ¢ g; (DiffOper) INTRODUCTION TO MANIFOLDS | III 3 is a Lie derivative along a certain vector ¯eld v 2 X. Idea of the proof. In local coordinates any function can be written as Xn @f f(x) = f(a) + (xk ¡ ak)fk(x); fk(a) = (a): @xk k=1 Applying the Leibnitz identity, we conclude that D = Lv, where v is the vector ¯eld with components vk = D(xk ¡ ak). ¤ Thus sometimes the notation Xn @ v = vk(x) @xk k=1 is used: such a notation understood as a di®erential oper- ator, is a vector ¯eld from the geometric point of view. 4. Commutator. If v; w 2 X(M), then D = LvLw ¡ LwLv is a di®erential operator. Indeed, the Leibnitz formula is trivially satis¯ed, therefore D = Lu, where u 2 X(M). | Problem 5. Check it! ~ De¯nition. If Lu = LvLw ¡ LwLv, then u is a commutator of v and w: u = [v; w]: In coordinates: à ! X @f Luf = Lv wk ¡ Lw(¢ ¢ ¢) = @xk k µ ¶ X 2 @ f @f @wk wkvj + vj ¡ (¢ ¢ ¢) = @xk@xj @xk @xj k;j à ! X X X @wj @vj @f vk ¡ wk ; @xk @xk @xj j k k therefore à ! X X X @wj @vj @ [v; w] = vk ¡ wk : @xk @xk @xj j k k | Problem 6. ¯ 2 ¯ @ ¯ t s s t ¯ (f ± v ± w ¡ f ± w ± v ) = L[v;w]f: @s@t s=0;t=0 | Problem 7. [v; w] = ¡[w; v]: | Problem 8. Prove the Jacobi identity [[u; v]; w] + [[v; w]; u] + [[w; u]; v] = 0: 4 ALGEBRA OF VECTOR FIELDS. LIE DERIVATIVE(S). 5. Lie derivation of vector ¯elds. ~ De¯nition. The Lie derivative of a vector ¯eld w along another ¯eld v is 1 t t Lvw = lim (v¤w ¡ w ± v ): t!0 t | Problem 9. Check that the above de¯nition makes sense. Properties of the Lie derivative: if v; w 2 X(M); f 2 C1(M), then: (1) Lvv = 0. (2) Lv is linear map from X(M) to itself. (3) Lv(fw) = (Lvf)w + f Lvw (the Leibnitz property). Theorem. Lvw = [v; w] (or [w; v]?) Proof. Let ¯ 2 ¯ @ ¯ t s s t a = ¯ (f ± v ± w ¡ f ± w ± v ): @s@t s=0;t=0 Then µ ¯ ¶ 1 @ ¯ a = lim ¯ (¢ ¢ ¢) ; t!0 ¯ t @s s=0 but ¯ ¯ @ ¯ t s t ¯ v ± w = v¤w; @s s=0 therefore ¯ ¯ @ t s ¯ f ± v ± w = L t f; ¯ v¤w @s s=0 while ¯ ¯ @ ¯ s t ¯ f ± w ± v = Lw±vt f; @s s=0 and ¯nally a = LLv wf: ¤ | Problem 10. Is the Lie derivative of a vector ¯eld local in the following sense: if two ¯elds v1; v2 2 X(M) are coinciding on an open neighborhood of a certain point a 2 M, then for any other ¯eld v 2 X(M) (Lv1 w)(a) = (Lv2 w)(a): Is it true that the above value is determined by the (common) value vi(a)? /black/users2/yakov/pub, ¯lename 4.ppt E-mail address: [email protected], [email protected].