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Lie Derivatives on Manifolds William C. Schulz 1. INTRODUCTION 2

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Lie Derivatives on Manifolds William C. Schulz 1. INTRODUCTION 2 Lie Derivatives on Manifolds William C. Schulz Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ 86011 1. INTRODUCTION This module gives a brief introduction to Lie derivatives and how they act on various geometric objects. The principal difficulty in taking derivatives of sections of vector bundles is that the there is no cannonical way of comparing values of sections over different points. In general this problem is handled by installing a connection on the manifold, but if one has a tangent vector field then its flow provides another method of identifying the points in nearby fibres, and thus provides a method (which of course depends on the vector field) of taking derivatives of sections in any vector bundle. In this module we develop this theory. 1 2. TANGENT VECTOR FIELDS A tangent vector field is simply a section of the tangent bundle. For our purposes here we regard the section as defined on the entire manifold M. We can always arrange this by extending a section defined on an open set U of M by 0, after some appropriate smoothing. However, since we are going to be concerned with the flow generated by the tangent vector field we do need it to be defined on all of M. The objects in the T (M) are defined to be first order linear operators acting ∞ on the sheaf of C functions on the manifold. Xp(f) is a real number (or a complex number for complex manifolds). At each point p ∈ M we have Xp(f + g) = Xp(f)+ Xp(g) Xp(fg) = Xp(f)g(p)+ f(p)Xp(g) Xp(f) should be thought of as the directional derivative of F in the direction X at P . Thus X inputs a function (at p) and outputs a real (or complex) number. We regard f as being defined on some neighborhood of p (which depends on f). The rules above describe how Xp acts on sums and products. It is possible to show, although we will not do so here, that Tp(M) is an n-dimensional vector space (n = dim M, the dimension of the Manifold M) ∂ ∂ with a basis in local coordinates { ∂u1 ,..., ∂un }. Hence i ∂ Xp = X (p) ∂ui p 17 Oct 2011 1 and i ∂f Xp(f)= X (p) ∂ui p i j If we change coordinates from u tou ˜ the the local expression of X changes like a contravariant tensor: ∂f ∂f ∂uj ∂f X˜ i = X˜ i = Xj ∂u˜i ∂uj ∂u˜i ∂uj so ∂uj ∂u˜i Xj = X˜ i and X˜ i = Xj ∂u˜i ∂uj 3. THE LIE BRACKET If we attempt to compose X and Y the results are not encouraging; locally ∂f Y (f) = Y i ∂ui ∂ ∂f X(Y (f)) = Xj Y i ∂uj ∂ui ∂Y i ∂f ∂2f = Xj + XjY i ∂uj ∂ui ∂uj ∂ui which shows that XY is not a tangent vector since it contains the second deriva- tive of f and is thus not a first order operator. However, we take heart from the observation that the objectionable term is symmetric in i and j. Thus if we form ∂Xi ∂f ∂2f Y (X(f)) = Y j + Y j Xi ∂uj ∂ui ∂uj∂ui and subtract, the objectionable terms will drop out and we have ∂Y i ∂Xi ∂f X(Y (f)) − Y (X(f)) = Xj − Y j ∂uj ∂uj ∂ui which is a first order operator and hence a tangent vector. We now introduce new notation ∂Y i ∂Xi ∂ [X, Y ] = X(Y (·)) − Y (X(·)) = Xj − Y j ∂uj ∂uj ∂ui ∂Y i ∂Xi [X, Y ]i = Xj − Y j ∂uj ∂uj For practise the user may wish to verify that [X, Y ]i transforms properly under coordinate change. ∂ ∂ X and Y are said to commute if [X, Y ] = 0. For example ∂ui and ∂uj commute. It is obvious that [X, Y ] is linear in each variable. 2 Next we form [X, [Y,Z]](f) = X((YZ − ZY )(f)) − (YZ − ZY )(X(f)) = X(Y (Z(f))) − X(Z(Y (f))) − Y (Z(X(f))) + Z(Y (X(f))) Let us abbreviate this by surpressing the f and the parentheses and then per- mute cyclically. [X, [Y,Z]] = XYZ − XZY − YZX + ZYX [Y, [Z,X]] = YZX − YXZ − ZXY + XZY [Z, [X, Y ]] = ZXY − ZYX − XYZ + YXZ Now if we add the three equations the terms cancel in pairs and we have the important Jacobi Identity [X, [Y,Z]] + [Y, [Z,X]] + [Z, [X, Y ]]=0 Thus the vector fields in T (M) form an (infinite dimensional) Lie Algebra, which is a vector space with a skew symmetric multiplication [X, Y ] linear in each variable satisfying the previous identity. The Jacobi identity substitutes for associativity. 4. BACK AND FORTH WITH DIFFEOMOR- PHISMS The user may wonder why this area of mathematics tends to emphasize diffeo- morphisms instead of differentiable maps. In fact, the level of generality implied by φ : M → N is largely spurious; most of the time N = M and so diffeomor- phisms are the natural objects to study. Nevertheless we follow convention in this regard and formulate the results for φ : M → N The user will recall that a mapping φ : M → N induces a mapping φ∗ = dφ from Tp(M) to Tq(N) where q = φ(p). Since a vector in Tq(N) may be identified by its action on a function f : V → R, V a neighborhood of q in N, we can define φ∗ by (φ∗X)q(f)= Xp(f ◦ φ) since f ◦ φ : M → R. For notational amusement we define φ∗f = f ◦ φ and then we have ∗ (φ∗X)(f)= X(φ f) and thus ∗ φ∗(X)= X ◦ φ 3 for X ∈ Tp(M) and φ∗ : Tp(M) → Tφ(p)(N). In local coordinates we describe φ as follows: p has coodinates u1,...,un and φ(p) has coodinates v1,...,vn and thus p → φ(p) is given by vi(u1,...,un) i =1,...,n Then, if ∂ ∂ X = Xi Y = Y i ∂ui ∂vj and Yφ(p) = φ∗(Xp) we have, (surpressing some p subscripts) ∂f ∂f ◦ φ ∂ui Y (f)= Y j = Y j φ(p) ∂vj ∂ui ∂vj and ∂f ◦ φ Y (f)= X (f ◦ φ)= Xi φ(p) p ∂ui from which we see that ∂ui Xi = Y j ∂vj and symmetrically ∂vj Y j = Xi ∂ui which resembles the coordinate change rules. This is for the best of reasons; because φ is a diffeomorphism a coordinate patch on N becomes, via φ, a coordinate patch on M. Thus, locally, a diffeomorphism looks like a coordinate change. This is not particularly helpful in keeping things straight in ones mind, although occasionally technically useful. Now if φ were just a differentiable mapping then it could not be used to map vector fields on M to vector fields on N. The obvious way to do this is as follows: given q ∈ N, select p ∈ M so that φ(p) = q and then let Yq = φ∗Xp. This won’t work for two reasons. First, there might be q ∈ N not in the range of φ, and even if q is in the range of φ we might have φ(p1) = φ(p2) = q for p1 6= p2 so that Yq would not be uniquely defined. However, neither of these is a problem for diffeomorphisms so that Def If X is a vector field on M and φ : M → N is a diffeomormorphism then we can define a vector field Y = φ∗(X) on N by (φ∗X)q = Xφ−1q ◦ φ = φ∗(Xφ−1q) If f : N → R then (recall φ∗(f)= f ◦ φ) ∗ (φ∗X)q(f) = Xφ−1q ◦ φ (f) = Xφ−1q(φ f) = Xφ−1q(f ◦ φ) Expressed slightly differently if Y = φ∗X then Yq(f) = Xφ−1q(f ◦ φ) 4 Notice that if φ1 : M1 → M2 and φ2 : M2 → M3 are diffeomorphisms then φ2∗ ◦ φ1∗ = (φ2 ◦ φ1)∗ This is just an abstract expression of the chain rule, but we can say it really fancy: ∗ is a covariant functor from the category of Manifolds and Diffeomor- phisms to the category of vector bundles and isomorphisms. Now we want to show that the Lie Bracket is preserved under diffeomor- phisms. First we note that if φ : M → N is a diffeomorphism then it can be regarded locally as a coordinate change. Those persons who verified that the Lie Bracket was invariant under coordinate change when I requested them to do so need not read the following. Let φ : M → N be a diffeomorphism and let u1,...,un be coordinates around p ∈ M and v1,...,vn be coordinates around q = φ(p) ∈ N. Then we have, for f : N → R, (φ∗X)qf = Xφ−1q(f ◦ φ) i ∂ = X i j (f ◦ φ) u (v ) ∂ui j i ∂f ∂v j ∂f = X i j = X˜ u (v ) ∂vi ∂ui vk ∂vj where we set j j i ∂v X˜ = X ℓ k vk u (v ) ∂ui Now we can calculate the Lie Bracket. We surpress the subscripts on X, X,Y,˜ Y˜ because they are always the same. Then ∂Y˜ i ∂X˜ i [X,˜ Y˜ ]i = X˜ j − Y˜ j ∂vj ∂vj ∂vj ∂ ∂vi ∂uℓ ∂vj ∂ ∂vi ∂uℓ = Xk Y m − Y k Xm ∂uk ∂uℓ ∂um ∂vj ∂uk ∂uℓ ∂um ∂vj ∂Y m ∂uℓ ∂vj ∂vi ∂vj ∂2vi ∂vℓ = Xk + XkY m ∂uℓ ∂vj ∂uk ∂um ∂uk ∂ui∂um ∂uj ∂Xm ∂uℓ ∂vj ∂vi ∂vj ∂2vi ∂vℓ − Y k − Y kXm ∂uℓ ∂vj ∂uk ∂um ∂uk ∂ui∂um ∂uj ∂Y m ∂vi ∂Xm ∂vi = Xk δℓ − Y k δℓ ∂uℓ k ∂um ∂uℓ k ∂um ∂2vi ∂2vi + XkY mδℓ − Y kXmδℓ k ∂uℓ∂um k ∂uℓ∂um ∂Y m ∂Xm ∂vi ∂2vi ∂2vi = Xℓ − Y ℓ + XkY m − Y kXm ∂uℓ ∂uℓ ∂um ∂uk∂um ∂uk∂um ∂vi = [X, Y ]m +0 ∂um ^ i = [X, Y ] 5 5.
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