Worksheet on Vector Fields and the Lie Bracket

Transcription

1 Worksheet on Vector Fields and the Lie Bracket Math 6456 Differential Geometry January 8, 2008 Let M be a differentiable manifold and T M := p M T p M Definition 8 A smooth vector field is a mapping w : M T M such that p w p T p M, and for every f F(M), the mapping p w p [f] is in F(M) Exercise 1 Given a local coordinate ξ : V R n with p V, the coordinate basis vectors v 1,, v n form a basis for T p M (we showed this in the last lecture) Show that when an arbitrary w p T p M is expressed as this defines functions w 1,, w n F(V ) w p = w j v j, (1) Exercise 2 Show that a mapping w : M T p M such that p w p T p M is smooth (ie, represents a smooth locally defined vector field) if and only if for every local coordinate ξ : V R n the coefficients w j in the representation (1) are (locally) in F(V ) Note: We are only considering smooth vector fields; when we refer to a vector field, it is always assumed to be smooth Exercise 3 Let V be open in M and p V Show that V is a differentiable manifold Note: T M sometimes refers to a different (equivalent) set: T M = {(p, v) : p M, v T p M} This collection of all tangent vectors on M (identified with their base points) is also called the tangent bundle Exercise 4 Show that the tangent bundle T M is a differentiable manifold of dimension 2n 1

2 The collection of all smooth vector fields on M is denoted by X(M) Exercise 5 1 Given a vector field w X(M), show that f w[f] defines a linear operator on F(M) Is this operator onto? (The space of all linear operators on the ring F(M) is sometimes denoted by L(F(M))) 2 Show that X(M) is isomorphic to a subspace (or a submodule) of L(F(M)) when considered as a vector space over R (or a module over F(M)) 3 Given v, w X(M), show that u p [f] := w p [v[f]] defines a linear map on F(M), ie, Λ(f) = u[f] F(M) and Λ L(F(M)) 4 find conditions under which u p [f] := w p [v[f]] defines a vector field u (When we write v[f] without any subscript p, we are considering the image of f under the image of v under isomorphism discussed in 1 and 2 above, ie, v[f] F) Hint: In light of 3 above, the only thing to check is the Leibnizian condition Show that u p [fg] = u p [f]g(p) + w p [f]v p [g] + v p [f]w p [g] + f[p]u p [g] Definition 10 If v, w X(M), the Lie bracket, [v, w] : F(M) F(M), of w with respect to v is defined by [v, w] p [f] := v p [w[f]] w p [v[f]] Exercise 6 Show that [v, w] X(M) Exercise 7 1 Let u p be the functional defined in Exercise 5, and compute an expression for u p in local coordinates 2 Compute an expression for [v, w] in local coordinates Show that this can be used as an approach to Exercise 6 Definition 11 A linear operator D L(F(M)) is a derivation if D(fg) = D(f)g + fd(g) Given v X(M), the Lie derivative L v : F(M) X(M) F(M) X(M) is defined by L v (f) = v[f] L v (w) = [v, w] if f F(M), if w X(M) Exercise 8 Show that L v (fw) = L v (f)w + fl v (w) 2

3 This kind of packaging together of different (Leibnizian/product related) operators on different spaces is typical of Riemannian geometry Exercise 9 Calculate L v iv j where v 1,, v n are the natural coordinate basis vectors (Hint: Consider the case of a surface in R 3 and recall that v 1,, v n correspond to the tangent vectors X 1, X 2 R 3 What vector in R 3 would correspond to L v iv j?) Exercise 10 If u, v, w X(M), a, b r, and f, g F(M), then 1 [v, w] = [w, v] This is called anticommutativity 2 [av + bw, u] = a[v, u] + b[w, u] This, of course, is linearity (and holds in both slots) 3 [[u, v], w] + [[w, u], v] + [[v, w], u] = 0 This is known as the Jacobi identity 4 [fv, gw] = fg[v, w] + fv[g]w + gw[f]v This is a kind of Leibniz identity The Flow Associated to a Vector Field In order to describe an interpretation of [v, w] as a derivative of the vector field w with respect to the vector field v, we consider a number of topics which are important in their own right Note that any open interval in the real line is a one-dimensional differentiable manifold with the identity map as global chart Let us, for the moment, use the notation I to denote this kind of particular manifold Definition 12 (smooth curve) A smooth map γ : I M is called a smooth curve in M The derivative or tangent vector of such a curve is defined by γ (t)[f] := d(f γ) (t) dt Notice that this is the same as d(f γ id 1 )/dt where id is the identity chart for I A smooth curve is called regular if γ 0 Exercise 11 Let v X(M) be a smooth vector field and p M Show that for some ɛ > 0, there is a smooth curve γ : ( ɛ, ɛ) M such that γ (t) = v γ(t) and γ(0) = p Show that γ is unique in the sense that if α : ( δ, δ) M satisfies α (t) = v α(t) and α(0) = p, then γ(t) = α(t) for t ( min{ɛ, δ}, min{ɛ, δ}) Hint: Use the existence and uniqueness theorem for ordinary differential equations 3

4 Definition 13 (integral curve) Given v X(M) and γ : I M any function defined on an open interval I We say that γ is an integral curve of v if γ (t) = v γ(t) The vector field v X(M) is called complete if every integral curve γ of v may be extended to the interval I = R In this case, we define the flow associated with v to be the mapping Φ : M R M satisfying 1 Φ(p, 0) p 2 Φ t (p, t) = v Φ(p,t) On the left of this equation, we are interpreting the partial derivative as the tangent vector to the curve t Φ(p, t) Exercise 12 Show that given any vector field v X(M) and a point p M, there is a complete vector field v 0 X(M) which agrees with v in a neighborhood of p In light of the last exercise, we will not hesitate to talk about the flow of any smooth vector field as long as we restrict attention to the locality of a point Exercise 13 Let v X(M) and let ξ : V R n be a local coordinate at p 1 Show that dξ may be used to define a smooth (classical) vector field w on U = ξ(v ) Hint: w(x) w x = dξ ξ 1 (x)(v ξ 1 (x)) 2 Consider the ordinary differential equation { ẋ = wx x(0) = x 0 for x 0 U Apply the existence and uniqueness theorem from ordinary differential equations to show the existence of a local flow for w 3 What is the relation between the flow of w on U and the flow of v on M? Exercise 14 Notice that M R is a manifold Show that the flow Φ of a complete vector field is a smooth map Exercise 15 Given any function f F(M) and any vector field v X(M) with local flow Φ at p M show that f Φ(p, t) f(p) lim = v p [f] h 0 t 4

5 At this stage, we can describe the Lie derivative of w with respect to v as follows: Let p be a point in M and take a complete vector field which agrees with v in a neighborhood of p Then we have a flow Φ and we can consider the (partial) differential map dφ : T M R T M which we get by thinking of t as fixed and taking the differential of the mapping p Φ(p, t) Exercise 16 Show that dφ is a smooth map If we wish to compare w p T p M and w Φ(p,t) T Φ(p,t) M, and we do, then we have some trouble since they are in different tangent spaces As a compromise, we will transfer w p over into T Φ(p,t) using the differential map; then we can form the difference quotient w Φ(p,t) dφ p (w p ) t The limit of this vector as t 0 is an alternative definition of the Lie derivative: w Φ(p,t) dφ p (w p ) L v w = lim To verify that this is correct, we need to see that this limit gives [v, w] = vw wv Let us start by applying the difference quotient to a function, using the definition of the differential, and using the linearity of w: 1 t (w Φ(p,t)[f] dφ p (w p )[f]) = 1 t (w Φ(p,t)[f] w p [f Φ]) = 1 t (w Φ(p,t)[f] w p [f]) + 1 t (w p[f] w p [f Φ]) = 1 t (w Φ(p,t)[f] w p [f]) w p [ 1 t (f Φ f) ] Of course, there is t dependence in the expression f Φ which we will have to deal with later, but notice that the first term has as limit w Φ(p,t) [f] w p [f] lim = t w Φ(p,t)[f] t=0 = Φ (p, 0)[w[f]] t = v p [w[f]] by definitions 12 and 13 For the same reason, it is clear that f Φ f lim = v[f] 5

6 at least pointwise The convergence of the interchanged limits 1 to compute [ ] f Φ f lim w p must, however, be justified We will address this in a more general context later, but for now, let us simply expand in local coordinates and argue directly: Lemma 6 [ ] f Φ f lim w p = w p [v[f]] Proof: Taking a local coordinate ξ about p and setting g(x, t) = f Φ(ξ 1 (x), t) on U R, we have [ ] f Φ f { } lim w p = lim wp [ξ j f Φ(ξ 1, t) f Φ(ξ 1, 0) ] t 0 x j t x=ξ(p) = lim wp [ξ j ] 1 ( g (x, t) g ) (x, 0) x j x j x=ξ(p) = w p [ξ j 2 g ] lim (x, t j t 0 t x ) j = w p [ξ j 2 g ] (x, 0) t x j where we have used the mean value theorem so that t j On the other hand, we may write is on the interval between 0 and t f Φ(ξ 1, h) f ξ 1 v ξ 1[f] = lim h 0 h 1 This interpretation of the Lie bracket may be found in do Carmo s Riemannian Geometry, but the overall proof is more complicated, while the issue of interchange of limits is ignored If we are to accept do Carmo s assertion that w p [g(, t)] w p [g(, 0)] as t 0, then our proof would simply read (1/t)(w Φ(p,t) [f] dφ p (w p )[f]) = (1/t)(w Φ(p,t) [f] w p [f]) w p [(1/t)(f Φ f)] v p [w[f]] w p [v[f]] 6

7 Therefore, This establishes the lemma w p [v[f]] = w p [ξ j ] v ξ 1[f] x j = w p [ξ j f Φ(ξ 1, h) f ξ 1 ] lim x j h 0 h = w p [ξ j ] lim x j t (x, h j ) = w p [ξ j g ] x j t (x, 0) = w p [ξ j 2 g ] x j t (x, 0) h 0 g x=ξ(p) Corollary 7 w Φ(p,t) dφ p (w p ) [v, w] = lim 7