Some jet bundle structure Andrew D. Lewis∗ 26/10/2005

Abstract Two results concerning the structure of jet bundles are presented and explained.

Introduction. We present, discuss, and hopefully make clear, two results stated in [Pom- maret 1978] as Lemma 9.12 and Proposition 9.13 in Chapter 1. The setup is a fibred manifold π : Y → X. By Vπ = ker(T π) we denote the vertical subbundle of TY. Let us denote ν = π ◦ (πTY|Vπ) the natural projection from Vπ to X. (By πTM : TM → M we denote the tangent bundle projection.) We denote by Jkπ the bundle of k k-jets of sections of Y, and we denote by πk : Jkπ → X and πl : Jkπ → Jlπ (k ≥ l) the natural projections. If (ξ, U) is a local section, then jkξ : U → Jkπ denotes the corresponding local section of Jkπ.

First result. We consider the fibred manifold ν : Vπ → X and denote by Jkν the bundle of k-jets of sections. We first make an observation about this manifold being, in fact, the total space of a vector bundle over Jkπ. Let us define the projection, which we denote by αk : Jkν → Jkπ. For a local section (η, U) of ν : Vπ → X, denote by (ξη, U) the local section of π : Y → X defined by ξη(x) = πTY(η(x)). Let us define αk(jkη(x)) = jkξη(x). We then have the following result.

1 Lemma: (The vector bundle structure of Jkν) The map αk : Jkν → Jkπ is a surjective submersion, and resulting fibred manifold has a vector bundle structure.

Proof: That αk is a surjective submersion is easily checked in local coordinates. We exhibit the vector bundle structure of αk : Jkν → Jkπ by indicating how one defines vector addition and scalar multiplication in the fibres of αk. Let (η1, U) and (η2, U) be local sections of

ν : Y → X such that jkξη1 (x) = jkξη2 (x) for all x ∈ U. Thus jkη1(x) and jkη2(x) are in the same fiber of αk. Moreover, η1(x) and η2(x) are in the same fibre of πTY|Vπ for each x ∈ U. Thus we may define the local section (η1 +η2, U) by asking that (η1 +η2)(x) = η1(x)+η2(x), where addition on the right takes place in V π. We then define ξη1 (x)

(jkη1 + jkη2)(x) = jk(η1 + η2)(x), so defining addition in the fibres of αk. Scalar multiplication is defined, for a ∈ R, by a(jkη(x)) = jk(aη)(x), where (aη, U) is the local section defined by aη(x) = a(η(x)) with scalar multiplication on the right being that in Vξη(x)π. Now one should really verify that there is a vector bundle structure by defining a vector bundle atlas. This is done easily in natural coordinates, however. We leave the working out of the details of this to the reader.  ∗Associate Professor, Department of Mathematics and Statistics, Queen’s University, Kingston, ON K7L 3N6, Canada Email: [email protected], URL: http://penelope.mast.queensu.ca/~andrew/

1 2 A. D. Lewis

To get an understanding of the vector bundle structure, let us consider in coordinates n m the case when k = 1. We denote adapted coordinates for Y by (x, y) ∈ U × V ⊂ R × R . m Coordinates for Vπ are denoted by (x, y, v) where v ∈ R . Coordinates for J1ν are then n m denoted by (x, y, v, p1, p2) where pa ∈ L(R ; R ) for a ∈ {1, 2}. Thus p1 is to be thought of as the partial derivatives of y with respect to x, and p2 is to be thought of as the partial derivatives of v with respect to x. The local representative of α1 is then easily seen to be

(x, y, v, p1, p2) 7→ (x, y, p1).

Thus, imagining how this looks for arbitrary k, the fibres of αk are formed by v and its partial derivatives with respect to x. The vector bundle structure is obtained from the fact that v is to be thought of as a vertical vector, and so is an element of a vector space. Addition and scalar multiplication are then just addition and scalar multiplication of these vectors and their derivatives. Now we consider another vector bundle over Jkπ, namely the vertical bundle Vπk of the fibred manifold πk : Jkπ → X. This is obviously a vector bundle over Jkπ. The final construction preliminary to the statement of the result is the following. Let η : X → Vπ be a section of ν : Vπ → X with ξη the corresponding section of π : Y → X. Let x0 ∈ X, let U be a neighbourhood of x0, and define ση,x0 : U × I → Y such that 1. I ⊂ R is an interval for which 0 ∈ int(I),

2. ση,x0 (x, 0) = ξη(x) for all x ∈ U,

◦ 3. π ση,x0 (x, t) = x for all (x, t) ∈ U × I, and 4. d σ (x , t) = η(x ) ∈ V π. dt t=0 η,x0 0 0 ξη(x0)

Thus ση,x0 is a variation of ξη at x0 in the direction of η(x0). Define jkση,x0 : U × I → Jkπ to be such that x 7→ jkση,x0 (x, t) is the k-jet of the local section x 7→ ση,x0 (x, t). Thus −1 t 7→ jkση,x0 (x, t) is a curve in πk (x) for each x ∈ U, and so has a tangent vector field that is vertical with respect to the projection πk. We now have the following result. d 2 Proposition: (Vπk ' Jkν) The map jkη(x) 7→ dt t=0jkση,x(x) is an isomorphism of vector bundles over Jkπ. Proof: This is easily, but tediously, checked in local coordinates.  Let us illustrate the result in the case where k = 1. We use, as above, local coordinates for J1ν denoted by m n m n m (x, y, v, p1, p2) ∈ U × V × R × L(R ; R ) × L(R ; R ).

Local coordinates for Vπ1 are denoted by n m m n m (x, y, p1, v, A) ∈ U × V × L(R ; R ) × R × L(R ; R ). The vector bundle isomorphism of the proposition looks like

(x, y, v, p1, p2) 7→ (x, y, p1, v, p2).

The idea, roughly speaking, is that jkη(x) is mapped to the vertical vector consisting of the derivatives of the vertical part of η. Some jet bundle structure 3

Second result. The second result we state has essentially to do with fixed point sets for the symmetric group. So let us first consider some seemingly pointless constructions along these lines. For m ∈ N denote by Sm the group of bijections of {1, . . . , m}, i.e., the symmetric L group of order m. For k, l ∈ N, let us define group monomorphisms ιk : Sk → Sk+l and R ιl : Sl → Sk+l by (σ(j), j ∈ {1, . . . , k}, ιL(σ)(j) = k j, j ∈ {k + 1, . . . , k + l}, and (j, j ∈ {1, . . . , k}, ιR(σ)(j) = l σ(j − k) + k, j ∈ {k + 1, . . . , k + l}.

L R Thus image(ιk ) consists of the permutations of only the first k elements and image(ιl ) L L consists of the permutations of only the last l elements. Let us denote Sk = image(ιk ) and R R L R Sl = image(ιl ). Note that Sk ×Sl then consists of those permutations that independently permute the first k terms and the last l terms. If Φ: G × A → A is an action of a group G on a set A, let

Fix(G, A) = {x ∈ A | Φ(g, x) = x, g ∈ G} be the fixed point set. We now have the following result.

3 Lemma: (Fixed point sets for actions of the symmetric group) Let k, l ∈ N and let Φ: Sk+l × A → A be an action of the symmetric group on the set A. Then

L R L R Fix(Sk+l,A) = Fix(Sk × Sl ,A) ∩ Fix(Sk+1 × Sl−1,A).

L R Proof: If x ∈ Fix(Sk+l,A) then Φ(σ1, x) = x and Φ(σ2, x) = x for all σ1 ∈ Sk × Sl and L R σ2 ∈ Sk+1 × Sl−1. Thus

L R L R x ∈ Fix(Sk × Sl ,A) ∩ Fix(Sk+1 × Sl−1,A) and so L R L R Fix(Sk+l,A) ⊂ Fix(Sk × Sl ,A) ∩ Fix(Sk+1 × Sl−1,A).

To prove the opposite inclusion we note that, since every element of Sk+l is the product of a finite number of transpositions, it suffices to prove that if

L R L R x ∈ Fix(Sk × Sl ,A) ∩ Fix(Sk+1 × Sl−1,A) then Φ(σ, x) = x for every transposition σ ∈ Sk+l. Indeed, if σ ∈ Sk+l is arbitrary, then write σ = σ1 ◦ ··· ◦ σr for transpositions σ1, . . . , σr. Then, if Φ(σa, x) = x for each a ∈ {1, . . . , r}, we have

Φ(σ, x) = Φ(σ1 ◦ ··· ◦ σr, x) = Φ(σ1, Φ(σ2,..., Φ(σr, x))) = x.

Thus we let L R L R x ∈ Fix(Sk × Sl ,A) ∩ Fix(Sk+1 × Sl−1,A) 4 A. D. Lewis

L R L R and we let σ ∈ Sk+l be a transposition. If σ ∈ Sk × Sl or if σ ∈ Sk+1 × Sl−1 then we immediately have Φ(σ, x) = x. Thus we may as well suppose that

L R L R σ ∈ Sk+l \ ((Sk × Sl ) ∪ (Sk+1 × Sl−1)).

If σ is a transposition of the elements j1, j2 ∈ {1, . . . , k +l}, this means that we may assume that j1 ∈ {1, . . . , k + 1} and j2 ∈ {k + 2, . . . , k + l}. Thus ! 1 ··· j ··· k + 1 ··· j ··· k + l σ = 1 2 . 1 ··· j2 ··· k + 1 ··· j1 ··· k + l

Let us write σ = σ1 ◦ σ2 ◦ σ1 where ! 1 ··· j1 ··· k + 1 ··· j2 ··· k + l σ1 = 1 ··· k + 1 ··· j1 ··· j2 ··· k + l ! 1 ··· j1 ··· k + 1 ··· j2 ··· k + l σ2 = , 1 ··· j1 ··· j2 ··· k + 1 ··· k + l

L R noting that σ1 ∈ Sk+1 and σ2 ∈ Sl . Thus Φ(σ1, x) = x and Φ(σ2, x) = x and so

Φ(σ, x) = Φ(σ1, Φ(σ2, Φ(σ1, x))) = x, and so x ∈ Fix(Sk+l,A) since we have shown that x is fixed by every transposition.  Now let us proceed with what we are really interested in here. We first will find it useful to have the following characterisation of Jkπ as a submanifold of an iterated collection of k 1 first-order jet bundles. For k ∈ N, define J1π inductively as follows. Take J1π = J1π, k thinking of this as a fibred manifold over X. Then define J1π to be the bundle of 1-jets of k−1 sections of J1 π over X. k 4 Lemma: (Jkπ is a fibred submanifold of J1π) The map

jkξ(x) 7→ j1 ··· j1 ξ(x) | {z } k times

k is an embedding of Jkπ in J1π as a fibred submanifold over X.

Proof: This is easily, if a little tediously, verified in natural coordinates.  Let us understand the lemma in coordinates in the case when k = 2. We denote adapted n m coordinates for Y by (x, y) ∈ U × V ⊂ R × R . Coordinates for J1π are then denoted by

n m (x, y, p1) ∈ U × V × L(R ; R ), coordinates for J2π are denoted by

n m 2 n m (x, y, p1, p2) ∈ U × V × L(R ; R ) × Lsym(R ; R ),

2 and coordinates for J1 are denoted by

n m n m 2 n m (x, y, p1, q1, q2) ∈ U × V × L(R ; R ) × L(R ; R ) × L (R ; R ), Some jet bundle structure 5

n n m 2 n m recalling that L(R ; L(R ; R )) ' L (R ; R ) via the isomorphism ι2 given by

ι2(L)(u1, u2) = L(u1)(u2). A working through in local coordinates of the map in the lemma gives the local model for 2 the submanifold J2π of J1π as

n 2 n m o (x, y, p1, q1, q2) | q1 = p1, q2 ∈ Lsym(R ; R ) .

k Now, using this inclusion of Jkπ in J1π, we wish to characterise Jk+lπ for k, l ∈ N. k+l First note that we have Jk+lπ as a submanifold of J1 π. Now consider the fibred manifold πl : Jlπ → X. Denote by Jkπl the bundle of k-jets of this fibred manifold. Note that Jlπ is l a submanifold of J1π with inclusion given by

jlξ(x) 7→ j1 ··· j1 ξ(x). | {z } l times k Also, Jkπl is a submanifold of J1πl with inclusion

jkjlξ(x) = j1 ··· j1 jlξ(x). | {z } k times

Since Jlπ is a fibred submanifold over X of Jkπl, this gives Jkπl as a fibred submanifold of k+l 1 J1 π. With these inclusions in mind, we state the following result.

5 Proposition: (Characterisation of Jk+lπ) For k, l ∈ N, Jk+lπ = (Jkπl) ∩ (Jk+1πl−1). n m Idea of Proof: We work locally, and so let Y = U × V ⊂ R × R , taking X = U and π(x, y) = x. For r ∈ N let U be the r-fold Cartesian product of U with itself. Denote by C ∞(Ur, V) the set of smooth maps from Ur to V. Define an equivalence relation in ∞ r C (U , V) by saying that f 1 and f 2 are equivalent at x0 ∈ U if

1. f 1(x0,..., x0) = f 2(x0,..., x0) and if,

2. for each r ∈ {1, . . . , r} and for each j1, . . . , jr ∈ {1, . . . , r},

Dj1 ··· Djr f 1(x0,..., x0) = Dj1 ··· Djr f 2(x0,..., x0).

With a little effort, one can convince oneself that the union over x0 of the sets of equivalence r classes at x0 is in 1–1 correspondence with the local model for J1π. ∞ r Now define an action Φ˜ of Sr on C (U , V) by ˜ Φ(σ, f)(x1,..., xr) = f(xσ(1),..., xσ(r)) r and define an action Φ on the local model for J1π by Φ(σ, [f]) = [Φ(˜ σ, f)].

k+l L R k+1 One can then verify that Jkπl ⊂ J1 π is exactly given by Fix(Sk × Sl , J1 π). After this verification, the result follows immediately from Lemma 3.  1 We use here the fact that, if Z ⊂ Y is a fibred submanifold of a fibred manifold π : Y → X, then Jk(π|Z) is a fibred submanifold of Jkπ over X for each k ∈ N. 6 A. D. Lewis

References

Pommaret, J.-F. [1978] Systems of Nonlinear Partial Differential Equations and Lie Pseu- dogroups, number 14 in Mathematics and its Applications, Gordon & Breach Science Publishers, New York, ISBN 0-677-00270-X.