Lifting Distributions to Tangent and Jet Bundles∗ Andrew D

Lifting distributions to tangent and jet bundles∗ Andrew D. Lewis† 1998/05/19 Last updated: 2001/08/17

Abstract Two natural means are provided to lift a distribution from a manifold to its tangent bundle, and they are shown to agree if and only if the original distribution is integrable. Two special cases, the case when the manifold is the total space of a vector bundle, and the case when the manifold is the total space of a ﬁbration over R, are dealt with in particular. For the latter case, the two constructions interact with the aﬃne structure of the corresponding jet bundles in the “same” way.

Keywords. distributions, tangent bundles, jet bundles, connections.

AMS Subject Classiﬁcations (2020). 53B05, 58A20, 58A30.

1. Introduction

If one is provided with a distribution D on a manifold M, there are some natural criteria one might ask for a “lifted” distribution on TM. For example, three possible natural properties are:

L1. if rank(D) = m then the rank of the lifted distribution should be 2m;

L2. if the tangent vector field to a curve c on M lies in D, then the tangent vector field to the curve c0 on TM (c0 is itself the tangent vector field of c) should lie in the lifted distribution;

L3. if D is an Ehresmann connection on a locally trivial ﬁbre bundle π : M → B with total space M (thus TM = D ⊕ ker(T π)), then the lifted distribution should be an Ehresmann connection on T π : TM → TB.

In this paper we provide two ways of lifting a distribution D to TM, both of which satisfy the three stated properties. The two methods are genuinely diﬀerent in that they agree if and only if D is integrable. In the event that M is the total space of a ﬁbre bundle over R,1 then we give conditions under which the lift to TM can be restricted to a distribution on J 1M, the bundle of one-jets of sections (recalling that J 1M is naturally a subset of TM

∗Unpublished †Professor, Department of Mathematics and Statistics, Queen’s University, Kingston, ON K7L 3N6, Canada Email: [email protected], URL: http://www.mast.queensu.ca/~andrew/ Research performed while Postdoctoral Fellow at the University of Warwick. Research in part supported by EPSRC Grant GR/K 99893. 1To avoid confusion, it should be stated that this investigation of ﬁbrations over R is unrelated to the property L3 stated as desirable for lifted distributions.

1 2 A. D. Lewis in this case). We shall see that a compatibility condition must be imposed in order that D lift to J 1M. The basic constructions we present for lifting vector fields and one-forms are well-known. For example, complete lifts of vector fields are discussed by Crampin and Pirani [1986], and it is no enormous stretch to extend this to distributions. Also, de León,Marrero, and Mart´ın de Diego [1997] give a coordinate version of our lifting for a codistribution in Section3. However, it is precisely these two constructions which are different, and we are able to completely characterise this difference in Section4. A method for lifting connections on a fibre bundle to the tangent fibre bundle is given by Koláˇr,Michor, and Slovák[1993, §46.7]. However, it is not clear that their construction may be performed for distributions which are not connections. It is also interesting to note that the construction given by Koláˇr, Michor, and Slovákdiffers from both of the constructions we give. Moreover, even though certain elements of what we present are well-established, it is worth cataloguing the ideas methodically. In Section5 the case is considered when M is the total space of a vector bundle, and the distribution or codistribution define a linear connection on this bundle. We show that, as expected, the two associated lifts provide linear connections on the tangent bundle. As a specialisation, this gives two ways of lifting an affine connection on a manifold to the tangent bundle of the manifold. One of the lifted affine connections is known in the existing literature as the complete lift [see Yano and Ishihara 1973], but the other, to the author’s knowledge, is new. The constructions we discuss here appear in the treatment of mechanical systems with affine constraints. Indeed, this is the setting of [de León,Marrero, and Mart´ınde Diego 1997]. In this context it is interesting to note that although we give two generally different ways of lifting a distribution from M to TM, the constructions are equivalent insofar as their application to constrained mechanical systems is concerned. This is one way to view the content of Proposition 6.3. We shall adopt throughout the geometric and differential calculus notation of [Abraham, Marsden, and Ratiu 1988]. In particular, we work in the category of C ∞ Banach manifolds, but give finite-dimensional coordinate formulas where appropriate. Let us review some elements of our notation here. If E and F are Banach spaces we denote by L(E; F ) the continuous linear mappings from E to F . If U is open in E, we denote by Df(u) ∈ L(E; F ) the derivative of f : U → F at u ∈ U. By Df(u) · e we mean the evaluation of the derivative at e ∈ E. If E is further a product, say E = E1 × E2, then we write u = (u1, u2) and denote the partial derivatives by Dif(u1, u2) ∈ L(Ei; F ), i = 1, 2. If A: U → L(E; F ) and f ∈ F , then DA(u) · (e, f) means the derivative with respect to u in the direction of e of the function from U to F given by u 7→ A(u)·f (this resolves potential ambiguity which arises from the order of the arguments e and f). If S is a subset of a Banach space E we denote by ann(S) the subspace of E∗ which annihilates F . Similarly, if T is a subset of E∗ we denote by coann(T ) the subspace of E annihilated by T . If (U, φ) is a chart for a manifold M taking values in a Banach space E, we denote by U = φ(U) the open subset of E which parameterises U. We denote by Γ∞(V ) the set of smooth sections of a vector bundle with total space V . πTM : TM → M denotes the tangent bundle of a manifold M, and T f : TM → TN is the derivative of f : M → N. The Vk restriction of T f to TxM we denote by Txf.T (V ) denotes the bundle of exterior k-forms associated with a vector bundle with total space V . Lifting distributions to tangent and jet bundles 3

2. Lifting distributions

In this section we lift a distribution D on M to one on TM, and show that the lifted distribution has the properties L1–L3 presented in the introduction. The basic constructions for lifting vector fields which we present here are well-known (see [Crampin and Pirani 1986, §13.2]). ∞ Let X ∈ Γ (TM). For each x0 ∈ M there exists a neighbourhood U of x0 on which the flow t 7→ expX (t): U → M is defined and is a local diffeomorphism for t sufficiently small. Thus we may define a one-parameter family of local diffeomorphisms t 7→ {vx 7→ c Tx(expX (t))(vx)} from TU to TM for t sufficiently small. We then define a vector field X on TM by defining

c d X (vx) = Tx expX (t)(vx). dt t=0 Note that we do not require X to be complete to make sense of this deﬁnition. Xc is called the complete lift of X. It will be helpful to have a coordinate expression for Xc. 2.1 Proposition: If X : U → E is the principal part of the vector ﬁeld X in a chart (U, φ), then (u, e) 7→ (X(u), DX(u) · e) is the principal part of Xc in the natural chart (T U, T φ).

Proof: Let u0 ∈ U and select a neighbourhood V of u0 sufficiently small that expX (t): V → U is a local diffeomorphism for t sufficiently small. Let us write the local flow as (t, u) 7→ c F (u, t) for u ∈ V . The flow of X is then (t, (u, e)) 7→ (F (u, t), D1F (u, t) · e). Thus we compute c X (u0, e) = (D2F (u0, 0) · 1, D1D2F (u0, 0) · (e, 1)) where we use the equality of mixed partial derivatives. The result now follows since X(u0) = D2F (u0, 0). In finite-dimensions, the coordinate expression for Xc is

∂ ∂Xi ∂ Xc = Xi + vj ∂xi ∂xj ∂vi if Xi, i = 1, . . . , n, are the components of X in a local chart, and we denote coordinates for TM by (xi, vj). Note that we shall employ the summation convention throughout the paper when writing coordinate formulas in finite-dimensions. There is at least one more natural way of defining a vector field on TM given one on M. If X ∈ Γ∞(TM) we define a vector field Xv, the vertical lift of X, on TM by

v d X (vx) = (vx + tX(x)). dt t=0 In a natural chart for TM one readily veriﬁes that the principal part of Xv is given by

Xv(u, e) = (0, X(u)) if X is the principal part of X. In finite-dimensions we have ∂ Xv = Xi ∂vi 4 A. D. Lewis if Xi, i = 1, . . . , n, are the components of X. Given a distribution D on M, we can now define a subset DT of TTM whose fibre over vx ∈ TM is T c ∞ Dvx = {X (vx) | X ∈ Γ (D)}. This is in fact a subbundle of TTM. 2.2 Proposition: The subset DT of TTM is a distribution on TM. Proof: Let (U, φ) be a chart for M inducing natural charts (T U, T φ) for TM and (T T U, T T φ) for TTM. Since D is a distribution on M, this means that Dx splits in TxM for each x ∈ M. Therefore, if we choose the chart domain U sufficiently small, we may find a vector bundle chart (T U, ψ) for TM with the following properties:

1. ψ is a bijection onto U × F1 × F2 for Banach spaces F1 and F2;

2. ψ(Dx) = {φ(x)} × F1 × {0};

3. the overlap map from T φ(TU) = U × E to ψ(TU) = U × F1 × F2 has the form

h:(u, e) 7→ (u, A1(u) · e, A2(u) · e)

for smooth maps Ai : U → L(E; Fi), i = 1, 2. We shall write the inverse of the overlap map h as

−1 h :(u, f1, f2) 7→ (u, B1(u) · f1 + B2(u) · f2) for maps Bi : U → L(Fi; E), i = 1, 2. Note that ( 0, i 6= j, Ai(u) ◦ Bj(u) = (2.1) idFi , i = j. We claim that the map

H : ((u, e), (e1, e2)) 7→ ((u, e), (A1(u) · e1,A2(u) · e1,

A1(u) · e2 − A1(u) · (DB1(u) · (e, A1(u) · e1) + DB2(u) · (e, A2(u) · e1)),

A2(u) · e2 − A2(u) · (DB1(u) · (e, A1(u) · e1) + DB2(u) · (e, A2(u) · e1)))) (2.2) is a local vector bundle isomorphism from U × E × E × E to U × E × F1 × F2 × F1 × F2. That it is a local vector bundle mapping is clear. That it is an isomorphism may be veriﬁed by checking, using (2.1), that its inverse is

−1 H : ((u, e), (f11, f21, f12, f22)) 7→ ((u, e), (B1(u) · f11 + B2(u) · f21,

DB1(u) · (e, f11) + B1(u) · f12 + DB2(u) · (e, f21) + B2(u) · f22)). (2.3)

Next we claim that (T T U, H ◦ T T φ) is a vector bundle chart for TTM which is adapted to DT (thus showing DT to be a subbundle). First let us get a local description of DT in the chart (T T U, T T φ). The ﬁbre of D at u ∈ U is locally given by

Du = {(u, B1(u) · f1) | f1 ∈ F1}. Lifting distributions to tangent and jet bundles 5

Thus a section X of D will have the form

X(u) = (u, B1(u) · f(u))

T for some f : U → F1. Therefore, by Proposition 2.1 the ﬁbre of D at (u, e) is given by

T D(u,e) = {((u, e), (B1(u)·f(u), DB1(u)·(e, f(u))+B1(u)·(Df(u)·e))) | f : U → F1}. (2.4)

Note that the map Φ(u0,e0) which sends the map f : U → F1 to (f(u0), Df(u0)·e0) ∈ F1 ×F1 is surjective for each (u0, e0) ∈ U × E. Indeed, for (f11, f12) ∈ F1 × F1 let A ∈ L(E; F1) have the property that A(e0) = f12. Then we see that the image of f : u 7→ f11 + A(u − u0) under Φ(u0,e0) is (f11, f12). Therefore we may take

T D(u,e) = {((u, e), (B1(u) · f11, DB1(u) · (e, f11) + B1(u) · f12)) | f11, f12 ∈ F1}. (2.5)

With this local description of DT , we claim that

◦ T H T T φ(Dvx ) = {(u, e)} × F1 × {0} × F1 × {0} where (u, e) = T φ(vx). Substituting (2.5) into the expression (2.2) for H shows that

◦ T H T T φ(Dvx ) ⊂ {(u, e)} × F1 × {0} × F1 × {0} The opposite inclusion follows immediately from (2.5), the deﬁnition (2.3) of H−1, and the property (2.1) of the maps Ai, i = 1, 2, and Bi, i = 1, 2.

2.3 Remarks: 1. In ﬁnite-dimensions, the gist of the proof is as follows. Let i j k l (x , v , u , w ) be natural coordinates for TTM and let {B1,...,Bn} be a local basis of vector ﬁelds on M for which Dx = spanR(B1(x),...,Bm(x)). Then the change of coordinates ∂As (xi, vj, vk, wl) 7→ xi, vj,Bkur,Bl wr − Bl t vpBtuq r r s ∂xp q

T i for TTM is adapted to the distribution D where Bj, i = 1, . . . , n are the components i i of Bj, and Aj is the inverse matrix of Bj.

2. If we have a local basis of vector ﬁelds {Xa | a = 1, . . . , m}, for D, then the vector c v T ﬁelds {Xa,Xb | a, b = 1, . . . , m} form a local basis for D . 3. Note that if Dv ⊂ TTM is the vertical lift of all tangent vectors in D then Dv ⊂ DT .

4. Clearly we may inductively define the lift of D to the higher order tangent bundles T kM. Let us denote the distribution so lifted to T kM by DT k . • We define the lift of D to be the distribution DT . From Remark 2.3–2 we readily see that if rank(D) = m then rank(DT ) = 2m. Thus we have satisfied the condition L1 in the introduction. Let us look at the condition L2 which concerns curves lying in D lifting to curves lying in DT . In the statement of the following result we abuse notation slightly by thinking of c0 as both the tangent vector field along a curve c and as a curve in TM. By c00 we mean the tangent vector field to c0 when c0 is thought of as a curve in TM. 6 A. D. Lewis

0 0 2.4 Proposition: Let c:[a, b] → M be a curve with c (t) ∈ Dc(t) for t ∈ [a, b]. If c : t 7→ 00 T Ttc · 1 is its time-derivative which is a curve in TM, then c (t) ∈ Dc0(t) for t ∈ [a, b]. Proof: We use the notation of Proposition 2.2. A curve c whose tangent vector ﬁeld lies in D will locally have the form t 7→ c(t) ∈ U where

Dc(t) · 1 = B1(c(t)) · f(t)

0 for some curve t 7→ f(t) in F1. The local form for c is then t 7→ (c(t), Dc(t) · 1). The local tangent vector to this curve at (c(t), Dc(t) · 1) is (Dc(t) · 1, D2c(t) · (1, 1)). Note that

2 D c(t) · (1, 1) = DB1(c(t)) · (Dc(t) · 1, f(t)) + B1(c(t)) · (Df(t) · 1)

0 T By (2.5) this shows that the tangent vector to c lies in D . To see that DT satisfies the condition L3 we stated in the introduction, we prove the following result. Recall that an Ehresmann connection for a fibration π : M → B is a complement HM to the vertical subbundle VM = ker(T π). 2.5 Proposition: If HM is an Ehresmann connection on the locally trivial fibre bundle π : M → B then (HM)T is an Ehresmann connection on T π : TM → TB. Proof: Let (U, φ) be an adapted chart for π : M → B. We shall take U = φ(U) to have the form U1 × U2 ⊂ E1 × E2 where Ui is an open subset of a Banach space Ei, i = 1, 2. The chart being adapted implies that the local representative of the projection π is given by (u1, u2) 7→ u1. Obviously, the local form of the vertical subbundle VM = ker(T π) consists of those elements ((u1, u2), (0, e2)) in the local model for TM. An Ehresmann connection defines a complement HM to VM in TM so this locally is given by elements of the form

H(u1,u2)U = {((u1, u2), (e1,C(u1, u2) · e1)) | e1 ∈ E1} for some map C : U → L(E1; E2). Therefore a general section of HU has the form

(u1, u2) 7→ ((u1, u2), (e(u1, u2),C(u1, u2) · e(u1, u2))) (2.6) for some map e: U → E1. The adapted chart (U, φ) for π : M → B induces a natural chart (T U, T φ) for TM. We write coordinates in this chart as (u1, u2, e1, e2). The complete lift of a local section of HM of the form (2.6) is a local section of (HM)T given by

(u1, u2, e1, e2) 7→ ((u1, u2, e1, e2), (e(u1, u2),

C(u1, u2) · e(u1, u2), D1e(u1, u2) · e1 + D2e(u1, u2) · e2,

D1C(u1, u2) · (e1, e(u1, u2)) + C(u1, u2) · (D1e(u1, u2) · e1)+

D2C(u1, u2) · (e2, e(u1, u2)) + C(u1, u2) · (D2e(u1, u2) · e2))).

Thus, by reasoning like that by which we derived (2.5) from (2.4), we see that

(HU)T = ((u , u , e , e ), (e ,C(u , u ) · e , e , (u1,u2,e1,e2) 1 2 1 2 11 1 2 11 12

D1C(u1, u2) · (e1, e11) + D2C(u1, u2) · (e2, e11) + C(u1, u2) · e12)) | e11, e12 ∈ E1 . (2.7) Lifting distributions to tangent and jet bundles 7

Now note that the local representative of T π has the form (u1, u2, e1, e2) 7→ (u1, e1). Thus

V(u1,u2,e1,e2)T U = {((u1, u2, e1, e2), (0, e21, 0, e22)) | e21, e22 ∈ E2}. (2.8)

Comparing (2.7) and (2.8) we see that (HM)T ∩ VTM = {0}. It is also evident that TTM = (HM)T + VTM. Indeed we may write

(e11, e21, e12, e22) = e11,C(u1, u2) · e11, e12, D1C(u1, u2) · (e1, e11) + D2C(u1, u2) · (e2, e11) + C(u1, u2) · e12 + 0, e21 − C(u1, u2) · e11, 0, e22− D1C(u1, u2) · (e1, e11) − D2C(u1, u2) · (e2, e11) − C(u1, u2) · e12 where the ﬁrst term on the right of the equals sign is horizontal, and the second is vertical. In ﬁnite dimensions we denote adapted coordinates for π : M → B by (xa, yα), a = 1, . . . , m, α = 1, . . . , n − m. A connection HM then locally has a basis ∂ ∂ + Cα(x, y) , a = 1, . . . , m ∂xa a ∂yα

α T for some local matrix function Ca , α = 1, . . . , n − m, a = 1, . . . , m.(HM) then has the local basis ∂ ∂ + Cα(x, y) , a = 1, . . . , m ∂ua a ∂vα ∂ ∂ ∂Cα ∂Cα ∂ + Cα(x, y) + b uc + b vβ , b = 1, . . . , m ∂xb b ∂yα ∂xc ∂yβ ∂vα where we denote coordinates on TM by ((xa, yα), (ub, vβ)). We remark that this does not agree with the construction in [Koláˇr,Michor, and Slovák1993, §46.7]. However, the induced connection they give also satisfies Proposition 2.4; what is not clear is whether the procedure they give is applicable to distributions which are not connections.

3. Lifting codistributions

Next we look at the lifting of a codistribution on M (i.e., a subbundle of T ∗M) to a codistribution on TM. The details of much of what we say in this section follow like those in Section2, so our presentation is somewhat more abbreviated. However, there are some subtle notational changes, so we cannot afford to be too brief. ∞ ∗ If β ∈ Γ (T M) we may define a function fβ on TM by fβ(vx) = β(x) · vx. We define c the complete lift of β to be the one-form β (vx) = dfβ(vx) on TM. One readily verifies that the local representative of βc in a natural chart for TM is

βc(u, e) = ((u, e), (Dβ(u) · (·, e), β(u))) (3.1) where β is the principal part of β in the chart. We have used here the “place holder” notation which we will employ to advantage in the remainder of the paper. To explain, by 8 A. D. Lewis

Dβ(u) · (·, e) we mean that element of E∗ which is the derivative at u ∈ U of the R-valued function u 7→ β(u) · e for ﬁxed e ∈ E. In ﬁnite-dimensions (3.1) reads

∂β βc = i vidxj + β dvi. ∂xj i We may now associate to a codistribution Λ a subset ΛT of T ∗TM by

T c ∞ Λvx = {β (vx) | β ∈ Γ (Λ)}. We call ΛT the lift of Λ to TM. The following result asserts that this is in fact a subbundle. 3.1 Proposition: If Λ is a codistribution on M then ΛT is a codistribution on TM.

Proof: Suppose M is modelled on a Banach space E. Let (U, φ) be a chart for M with (T ∗U, T ∗φ) the induced natural chart for T ∗M. Since Λ is a subbundle of T ∗M we may choose U so that there exists a vector bundle chart (T ∗U, χ) for T ∗M with the following properties:

∗ ∗ 1. χ is a bijection onto U × V1 × V2 for Banach spaces V1 and V2; ∗ 2. χ(Λx) = {φ(x)} × {0} × V2 ; ∗ ∗ ∗ ∗ ∗ ∗ 3. the overlap map from T φ(T U) = U × E to χ(T U) = U × V1 × V2 is given by

g :(u, α) 7→ (u, P1(u) · α, P2(u) · α)

∗ ∗ for maps Pi : U → L(E ; Vi ), i = 1, 2. We denote the inverse of the overlap map by

−1 1 2 1 2 g :(u, ν , ν ) 7→ (u, Q1(u) · ν + Q2(u) · ν )

∗ ∗ for maps Qi : U → L(Vi ; E ), i = 1, 2. The relations ( 0, i 6= j Pi(u) ◦ Qj(u) = (3.2) id ∗ , i = j Vi hold. With respect to the chart (U, φ) for M we write natural coordinates for T ∗TM as ((u, e), (α1, α2)) ∈ U × E × E∗ × E∗. We claim that the map

1 2 1 G: ((u, e), (α , α )) 7→ ((u, e), (P1(u) · α − ∗ 2 ∗ 2 DQ1(u) · (P1 (u) · (·),P1(u) · α , e) − DQ2(u) · (P1 (u) · (·),P2(u) · α , e), 1 ∗ 2 P2(u) · α − DQ1(u) · (P2 (u) · (·),P1(u) · α , e)− ∗ 2 2 2 DQ2(u) · (P2 (u) · (·),P2(u) · α , e),P1(u) · α ,P2(u) · α )) (3.3)

∗ ∗ ∗ ∗ ∗ ∗ is a local vector bundle isomorphism from U × E × E × E to U × E × V1 × V2 × V1 × V2 which provides a chart for T ∗TM which is a subbundle chart for ΛT . Here, for example, Lifting distributions to tangent and jet bundles 9

∗ 2 ∗ by DQ1(u) · (P1 (u) · (·),P1(u) · α , e) we mean that element of V1 whose value at v1 ∈ V1 is deﬁned to be the derivative of the function

0 0 2 ∗ U 3 u 7→ (Q1(u ) ◦ P1(u) · α ) · e ∈ V1

∗ in the direction of P1 (u) · v1 ∈ E. That (3.3) deﬁnes a local vector bundle mapping is obvious. Furthermore, its inverse may be veriﬁed to be

−1 11 21 12 22 12 11 G : ((u, e), (ν , ν , ν , ν )) 7→ ((u, e), (DQ1(u) · (·, ν , e) + Q1(u) · ν + 22 21 12 22 DQ2(u) · (·, ν , e) + Q2(u) · ν ,Q1(u) · ν + Q2(u) · ν )), (3.4) using (3.2). Now we show that G provides a subbundle chart for ΛT . We need to compute the local form of sections of ΛT . In our local model (T ∗U, T ∗φ) for T ∗M a section of Λ has the form

u 7→ (u, Q2(u) · ν(u))

∗ for some ν : U → V2 . Therefore, using (3.1) and arguments very much like those used in the proof of Proposition 2.2 we derive

T 21 22 21 21 22 ∗ Λ(u,e) = {((u, e), (DQ2(u) · (·, ν , e) + Q2(u) · ν ,Q2(u) · ν )) | ν , ν ∈ V2 }. (3.5)

The relation T ∗ ∗ G(Λ(u,e)) ⊂ {(u, e)} × {0} × V2 × {0} × V2 is derived by substituting the form (3.5) for ΛT into the deﬁnition (3.3) for G. The opposite −1 inclusion follows directly from the deﬁnition (3.4) of G .

i j 3.2 Remarks: 1. In ﬁnite-dimensions the result shows that if (x , v , αk, βl) are natural coordinates for T ∗TM and Q1,...,Qn form a local basis for one-forms on M with m+1 n T Λx = spanR(Q (x),...,Q (x)), then adapted coordinates for Λ are given by ∂Qs xi, vj,P rα − P r t vtP qβ ,P rβ . k r k ∂xr s q l r

i i i i Here Qj, j = 1, . . . , n, are the components of Q , and Pj is the inverse matrix of Qj. 2. If {βm+1, . . . , βn} form a local basis of one-forms for Λ, then the one-forms a c ∗ b T {(β ) , πTM β | a, b = m + 1, . . . , n} form a local basis for Λ . ∗ ∗ T 3. Note that the subbundle πTM Λ of T TM is contained in Λ . 4. As we did with distributions, we can lift a codistribution Λ to the kth tangent bundle T kM. Let us denote the codistribution so lifted as ΛT k . • The following result further characterises ΛT . 10 A. D. Lewis

3.3 Proposition: Let Λ be a codistribution on M and let c:[a, b] → M be a curve on TM 0 00 T for which c (t) Λc(t) = 0 for t ∈ [a, b]. Then c (t) Λc0(t) = 0 for t ∈ [a, b]. Proof: We use the notation of Proposition 3.1. If c is the local representative of c then we must have 2 (Q2(c(t)) · ν ) · (Dc(t) · 1) = 0 2 ∗ 0 for every ν ∈ V2 since Λ must annihilate c . Differentiating this with respect to t gives 2 2 2 DQ2(c(t)) · (Dc(t) · 1, ν , Dc(t) · 1) + (Q2(c(t)) · ν ) · D c(t) · (1, 1) = 0 2 ∗ 0 for every ν ∈ V2 . By (3.5), and since the local representative of c is t 7→ (c(t), Dc(t) · 1) 00 T this means that c is annihilated by Λ as was desired. Remark 3.2–2 and Proposition 3.3 show that for the distribution coann(ΛT ) the properties L1 and L2 of the introduction hold. Now let us show that L3 holds as well. It is convenient to use an alternate, equivalent, definition of an Ehresmann connection from that used in Section2. Note that H∗M = ann(VM) is a naturally defined subbundle of T ∗M. Thus an Ehresmann connection may be regarded as a complement V ∗M to H∗M in TM. 3.4 Proposition: If V ∗M is a connection on π : M → B then the codistribution (V ∗M)T is a connection on T π : TM → TB. Proof: We must show that T ∗TM = H∗TM ⊕ (V ∗M)T . We shall take advantage of the coordinate notation introduced in the proof of Proposition 2.5. Thus we let (U, φ) be an adapted chart for π : M → B taking values in E1 × E2, and we recall that the horizontal subspace at (u1, u2) ∈ U is given by

H(u1,u2)U = {((u1, u2), (e1,C(u1, u2) · e1)) | e1 ∈ E1} ∗ for some C : U → L(E1; E2). One easily sees that the annihilating codistribution V M has the local representative V ∗ U = ((u , u ), (−C∗(u , u ) · α2, α2)) | α2 ∈ E∗ . (3.6) (u1,u2) 1 2 1 2 2 In the natural chart (T U, T φ) for TM the local representation (3.5) of (V ∗M)T gives

(V ∗U)T = ((u , u , e , e ), (−D C∗(u , u ) · (·, α22, e ) − C∗(u , u ) · α21, (u1,u2,e1,e2) 1 2 1 2 1 1 2 1 1 2 21 ∗ 22 ∗ 22 22 21 22 ∗ α , −D2C (u1, u2) · (·, α , e1), −C (u1, u2) · α , α )) | α , α ∈ E2 . (3.7) We also have H∗ TU = {((u , u , e , e ), (α11, 0, α12, 0)) | α11, α12 ∈ E∗}. (u1,u2,e1,e2) 1 2 1 2 1 We then easily see that H∗TM ∩ (V ∗M)T = {0}. We may also write

11 21 12 22 ∗ 22 ∗ 21 (α , α , α , α ) = −D1C (u1, u2) · (·, α , e1) − C (u1, u2) · α − ∗ 22 21 ∗ 22 22 D2C (u1, u2) · (C(u1, u2) · (·), α , e1), α , −C (u1, u2) · α , α + 11 ∗ 22 ∗ 21 α + D1C (u1, u2) · (·, α , e1) + C (u1, u2) · α + ∗ 22 12 ∗ 22 D2C (u1, u2) · (C(u1, u2) · (·), α , e1), 0, α + C (u1, u2) · α , 0 ∗ T ∗ where the ﬁrst term is in (V U) and the second is in H TU, which proves the result. Lifting distributions to tangent and jet bundles 11

Let us examine how this looks in finite-dimensions. We let (xa, yα), a = 1, . . . , m, α = 1, . . . , n−m, be coordinates in an adapted chart, with ((xa, yα), (ub, vβ)) the coordinates for the natural chart for TM. Then we may choose one-forms of the form α α a dy − Ca (x, y)dx , α = 1, . . . , n − m as local generators for V ∗M. The corresponding local generators for (V ∗M)T are then α α a dy − Ca (x, y)dx , α = 1, . . . , n − m ∂Cβ ∂Cβ dvβ − Cβ(x, y)dua − a uadxb − a uadyγ, β = 1, . . . , n − m. a ∂xb ∂yγ Note that, as with the construction of Section2, this differs from the formulas given in [Koláˇr,Michor, and Slovák1993, §46.7] for an induced connection on TM.

4. When the codistribution and distribution annihilate one another

In Sections2 and3 we gave two methods for lifting a distribution on M to one on TM, in the latter case the definition being made via the annihilating codistribution. It might be natural to expect that the two constructions are the same in that (ann(D))T = ann(DT ). Interestingly, however, this is generally not the case. In this section we let D be a distribution and denote by Λ its annihilating codistribution. First let us just look at the case of a single one-form and vector field which vanish when paired. When performing the exterior derivative in infinite-dimensions we use the characterisation in terms of the Lie derivative given by Palais [1954]. 4.1 Proposition: If β ∈ Γ∞(T ∗M) and X ∈ Γ∞(TM) have the property that β · X = 0, c c then β · X (vx) = dβ(X(x), vx) for all vx ∈ TxM and x ∈ M. Proof: We work locally with a chart (U, φ) for M with corresponding natural charts for TM, T ∗M, TTM, and T ∗TM. We suppose that β and X have local forms β(u) = (u, β(u)),X(u) = (u, X(u)). We then have βc(u, e) = ((u, e), (Dβ(u) · (·, e), β(u))) Xc(u, e) = ((u, e), (X(u), DX(u) · e)). Therefore, βc · Xc(u, e) = Dβ(u) · (X(u), e) + β · (DX(u) · e). (4.1) If we differentiate the expression β(u) · X(u) = 0 in the direction of e ∈ E we get Dβ(u) · (e, X(u)) + β(u) · (DX(u) · e) = 0 which, upon substitution into (4.1), gives βc · Xc(u, e) = Dβ(u) · (X(u), e) − Dβ(u) · (e, X(u)).

The right-hand side is exactly the local representation of dβ(X(x), vx). With this formula, we may easily compare ΛT with DT . While we are at it, let us also relate integrability of D to that of DT . 12 A. D. Lewis

4.2 Proposition: Let D be a distribution on M with Λ its annihilating codistribution. The following are equivalent: (i) ΛT = ann(DT ); (ii) D is integrable; (iii) DT is integrable. Proof: (i) ⇐⇒ (ii) Assume (i). Deﬁne 2 V2 Λ = {Ω ∈ T (TxM) | Ω(v1, v2) = 0 for all v1, v2 ∈ Dx and x ∈ M} and recall that D is integrable if and only if dβ ∈ Γ∞(Λ2) for all β ∈ Γ∞(Λ) [Abraham, Marsden, and Ratiu 1988, page 439]. If ΛT = ann(DT ) then βc ·Xc = 0 for every β ∈ Γ∞(Λ) ∞ and X ∈ Γ (D). By Proposition 4.1 this means that dβ(ux, vx) = 0 for every ux ∈ Dx and every vx ∈ TxM, and for every x ∈ M. In particular, D is integrable. Now suppose D integrable and let (U, φ) be a chart adapted to the foliation associated with D. Thus we can take U = U1 × U2 ⊂ E1 × E2 so that

D(u1,u2) = {((u1, u2), (e1, 0)) | e1 ∈ E1}. Thus we also have 2 2 ∗ Λ(u1,u2) = {((u1, u2), (0, α )) | α ∈ E2 }. One then readily ascertains that DT = {((u , u , e , e ), (e , 0, e , 0)) | e , e ∈ E }. (4.2) (u1,u2,e1,e2) 1 2 1 2 11 12 11 12 1 and ΛT = {((u , u , e , e ), (0, α21, 0, α22)) | α21, α22 ∈ E∗}. (u1,u2,e1,e2) 2 2 1 2 2 From this we immediately see that (i) must hold. (ii) ⇐⇒ (iii) Assume D integrable and let (U, φ) be a chart adapted to the foliation associated with D as above. From (4.2) it immediately follows that DT is integrable. Now suppose that DT is integrable. We shall employ the notation introduced in Propo- sition 2.2. Consider local sections of DT given by

X(u, e) = ((u, e), (B1(u) · f1, DB1(u) · (e, f1) + B1(u) · f2))

Y (u, e) = ((u, e), (B1(u) · f3, DB1(u) · (e, f3) + B1(u) · f4)) for f1, f2, f3, f4 ∈ F1. For the computations we perform here, it is suﬃcient to take these elements of F1 to be constant. A computation then gives

[X,Y ](u, 0) = ((u, 0), (DB1(u) · (B1(u) · f3, f1) − DB1(u) · (B1(u) · f1, f3),

DB1(u) · (B1(u) · f3, f2) − DB1(u) · (B1(u) · f2, f3)+

DB1(u) · (B1(u) · f4, f1) − DB1(u) · (B1(u) · f1, f4). Note that T D(u,0) = {((u, 0), (B1(u) · f5,B1(u) · f6)) | f5, f6 ∈ F1}. Therefore, since DT is integrable we must therefore have

DB1(u) · (B1(u) · f3, f1) − DB1(u) · (B1(u) · f1, f3) = B1(u) · f5 for some f5 ∈ F1. However, this is exactly the condition that Lie brackets of sections of D should remain sections of D—that is to say, D is integrable. Lifting distributions to tangent and jet bundles 13

5. When M is the total space of a vector bundle

In this section we specialise to the interesting case when M is the total space of a vector bundle, and we denote this vector bundle as π : E → B. An especially interesting, and well- studied, case of this occurs when the linear connection is derived from an aﬃne connection. We treat this in Section 5.4.

5.1. Linear connections. Suppose that we have a connection HE on π, and let ver: TE → VE be the projection onto the vertical subbundle. If ver is a vector bundle mapping with respect to the following diagram,

ver TE / VE

T π " | T π|VE TB then we say that HE is a linear connection. The following local characterisation of a linear connection will also be useful. We let (U, φ) be a vector bundle chart for E taking values in the Banach space E1 × E2. We regard E2 as the local model for the ﬁbres so that φ(U) = U × E2 for some open subset U of E1. As in the proof of Proposition 2.5, an Ehresmann connection is deﬁned locally by

H(u,e)U × E2 = {((u, e), (e1,C(u, e) · e1)) | e1 ∈ E1}, for some map C : U × E2 → L(E1; E2). One may verify that the Ehresmann connection is linear exactly when C has the form C(u, e) = Γ(u) · e for a map Γ: U → L(E2; L(E1; E2)). We call Γ the Christoffel map for the linear connection. We recall that if one has a linear connection HE on a vector bundle π : E → B then this ∞ defines, for each X ∈ Γ (TB), a derivation ∇X on the sections of E. Thus, for example, given a section σ : B → E, one can define a section ∇X σ. In local coordinates we have

∇X σ(u) = (u, Dσ(u) · X(u) + (Γ(u) · X(u)) · σ(u)).

In the case when E = TM, B = M, and π = πTM , this gives the usual definition of an affine connection on M. As a final trivial note, we observe that if π : E → B is a vector bundle then so too it T π : TE → TB. If (u, e) are adapted coordinates for E with ((u, e), (e1, e2)) natural coordinates for TM, then the coordinates ((u, e1), (e, e2)) are adapted for the vector bundle T π : TE → TB.

5.2. Lifting linear connections using distributions. The result in this section is straightforward and hopefully not unexpected. 5.1 Proposition: If HE is a linear connection on a vector bundle π : E → B then (HE)T is a linear connection on the vector bundle T π : TE → TB. Proof: We adopt the notation of Section 5.1 for the local representation of the linear connection HE. Thus the local model for HE is given by

H(u,e)(U × E2) = {((u, e), (e1, (Γ(u) · e) · e1)) | e1 ∈ E1}. (5.1) 14 A. D. Lewis

If we refer to the proof of Proposition 2.5, particularly to (2.7), we see that (HE)T has the local model

(H(U × E ))T = ((u, e, e , e ), (e , (Γ(u) · e) · e , e , 2 (u,e,e1,e2) 1 2 11 11 12

(DΓ(u) · (e1, e)) · e11 + (Γ(u) · e2) · e11 + (Γ(u) · e) · e12)) | e11, e12 ∈ E1 . (5.2)

Note that in this expression the local coordinates (u, e, e1, e2) that we use are the natural tangent bundle coordinates for TE associated with the vector bundle coordinates (u, e) for E. These are not adapted coordinates for the vector bundle T π : TE → TB; these would T be (u, e1, e, e2). If we write the local model for (HE) in these coordinates adapted to the vector bundle T π : TE → TB, then we have

(H(U × E ))T = ((u, e , e, e ), (e , e , (Γ(u) · e) · e , 2 (u,e1,e,e2) 1 2 11 12 11

(DΓ(u) · (e1, e)) · e11 + (Γ(u) · e2) · e11 + (Γ(u) · e) · e12)) | e11, e12 ∈ E1 .

Now we simply observe that this does indeed have the form of a linear connection in adapted coordinates as per (5.1).

5.2 Remarks: 1. In coordinates in finite-dimensions, suppose that we denote vector bundle coordinates for E by (xi, ua), and that the coordinates for TE adapted to the vector bundle T π : TE → TB are (xi, vj, ua, wb). The Christoffel symbols for a linear connection are defined in these coordinates by

∂ b ∂ ∇ ∂ a = Γia b , i = 1, . . . , n, a = 1, . . . , m. ∂xi ∂u ∂u In this case, if ∂ ∂ − Γa ub , i = 1, . . . , n, ∂xi ib ∂ua form a basis for HE, then ∂ ∂ − Γa ub , i = 1, . . . , n ∂vi ib ∂wa ∂ ∂ ∂Γa ∂ − Γa ub − jb ub + Γa wb , j = 1, . . . , n, ∂xj jb ∂ua ∂vk jb ∂wa form a basis for (HE)T .

2. Note that the lifted linear connection, as a distribution on TE, must be thought of as a connection with the proper base space, which is TB and not E. In the tangent bundle situation, TB = E = TM, so one expects to be able to establish a relationship between the two types of connections. We do this in Section 5.4. •

5.3. Lifting linear connections using codistributions. We may now mirror Proposition 5.1, except we now use codistributions to do the lifting. In this setting, of course, it is convenient to use the dual notion of a linear connection. Thus we say that a subbundle V ∗E of T ∗E is a linear connection if coann(V ∗E) is a linear connection in the sense of Section 5.1. Lifting distributions to tangent and jet bundles 15

5.3 Proposition: If V ∗E is a linear connection on a vector bundle π : E → B then (V ∗E)T is a linear connection on the vector bundle T π : TE → TB.

Proof: We adopt the notation of Proposition 5.1. Referring to the proof of Proposition 3.4, particularly to (3.6), we see that the local model for V ∗E has the form

∗ ∗ 2 2 2 ∗ V(u,e)(U × E2) = {((u, e), (−(Γ (u) · e) · α , α )) | α ∈ E2 }, (5.3)

∗ ∗ ∗ ∗ ∗ ∗ where Γ : U → L(E2; L(E2 ; E1 )) is speciﬁed by asking that Γ (u) · e ∈ L(E2 ; E1 ) be the dual endomorphism of Γ(u) · e ∈ L(E1; E2). We then make an application of (3.7) to ascertain that the local model for (V ∗E)T is

(V ∗(U × E ))T = ((u, e, e , e ), −((DΓ∗(u) · (·, e)) · α22) · e − (Γ∗(u) · e) · α21, 2 (u,e,e1,e2) 1 2 1 21 ∗ 22 ∗ 22 22 21 22 ∗ α − ((Γ (u) · (·)) · α ) · e1 − (Γ (u) · e) · α , α ) | α , α ∈ E2 .

The result now follows in the same way as did Proposition 5.1, after we use coordinates (u, e1, e, e2) that are adapted to the vector bundle T π : TE → TB. Indeed, writing the local model for (V ∗E)T in these coordinates yields

(V ∗(U × E ))T = ((u, e , e, e ), −((DΓ∗(u) · (·, e)) · α22) · e − (Γ∗(u) · e) · α21, 2 (u,e1,e,e2) 1 2 1 ∗ 22 ∗ 22 21 22 21 22 ∗ − ((Γ (u) · (·)) · α ) · e1 − (Γ (u) · e) · α , α , α ) | α , α ∈ E2 .

All one need do is observe that this has the form of (5.3).

5.4 Remarks: 1. Recall the coordinates used in Remark 5.2–1. If the one-forms

a a b i du + Γibu dx , a = 1, . . . , m,

form a basis for V ∗E, then the one-forms

a a c i du + Γicu dx , a = 1, . . . , m ∂Γb dwb + ic viucdxj + Γb viduc + Γb ucdvi, b = 1, . . . , m, ∂xj ic ic form a basis for (V ∗E)T .

2. Of course, Remark 5.2–2 holds equally well here. •

5.4. When the linear connection comes from an affine connection. It is now a simple matter to consider the results of Sections 5.2 and 5.3 when applied to the case when E = TM, B = M, and π = πTM . In this case, a linear connection HTM on πTM : TM → M is exactly specified by an affine connection ∇ on M. One would then like to lift this affine connection to an affine connection on TM using the two constructions we have provided. Note that a direct application of the two constructions (HTM)T and (V ∗TM)T will not give what is desired in this respect. These connections are on the bundle T πTM : TTM → TM, whereas an affine connection on TM will define a linear connection on πTTM : TTM → TM. 16 A. D. Lewis

To repair the discrepancy, we use the canonical involution of the double tangent bundle ΦM : TTM → TTM that is given in local coordinates by

ΦM ((u, e), (e1, e2)) = ((u, e1), (e, e2)).

Of course, an intrinsic definition of ΦM exists, but we shall not give it here. The appropriate manner in which to use the involution ΦM to modify the connections of the preceding two sections turns out to be as follows. For a distribution D on TTM we define a distribution ∗ ΦM D on TTM by declaring that the fibre at Xvx ∈ Tvx TQ be given by

∗ ∗ ∞ ΦM DXvx = {ΦM Z(Xvx ) | Z ∈ Γ (D)}.

∗ Thus ΦM D is deﬁned by declaring that its sections be pull-backs of sections of D by ΦM . Our main result in this section is the following.

5.5 Proposition: Let HTM be a linear connection on the bundle πTM : TM → M with V ∗TM = ann(HTM). The following statements hold: ∗ T (i) ΦM ((HTM) ) is a linear connection on the bundle πTTM : TTM → TM; ∗ ∗ T (ii) ΦM (coann((V TM) )) is a linear connection on the bundle πTTM : TTM → TM. Furthermore, the linear connections of (i) and (ii) agree if and only if the aﬃne connection on M associated with HTM is ﬂat.

Proof: Part (i) of the proposition will follow from Proposition 5.1 after we provide the local form of a linear connection on πTTM : TTM → TM. Let us use coordinates (u, e) ∈ U × E as natural coordinates for TM, noting then that the natural tangent bundle coordinates ((u, e), (e1, e2)) for TTM are adapted to the bundle πTTM : TTM → TM. Thus a linear connection on πTTM : TTM → TM will have the local form

H(TTM) = ((u, e, e1, e2), (e11, e12,

(Γ111(u, e) · e1) · e11 + (Γ121(u, e) · e2) · e11 + (Γ112(u, e) · e1) · e12+

(Γ122(u, e) · e2) · e12, (Γ211(u, e) · e1) · e11 + (Γ221(u, e) · e2) · e11+

(Γ212(u, e) · e1) · e12 + (Γ222(u, e) · e2) · e12)) | e11, e12 ∈ E , for appropriate functions Γabc : U × E → L(E; L(E; E)), a, b, c = 1, 2. For (i), a reference to (5.2) and the deﬁnition of pull-back shows that

Φ∗ (HTM)T = ((u, e, e , e ), (e , e , (Γ(u) · e ) · e , M (u,e,e1,e2) 1 2 11 12 1 11

(DΓ(u) · (e, e1)) · e11 + (Γ(u) · e2) · e11 + (Γ(u) · e1) · e12)) | e11, e12 ∈ E1 .

∗ T This shows that, indeed, ΦM ((HTM) ) is a linear connection on πTTM : TTM → TM, thus proving (i). In fact, the above computations are readily generalised to show that if ∗ D is a linear connection on T πTM : TTM → TM, then ΦM D is a linear connection on πTTM : TTM → TM. This directly implies (ii). The ﬁnal assertion of the proposition follows directly from Proposition 4.2, recalling that an aﬃne connection has zero curvature if and only if its associated linear connection is integrable [Kobayashi and Nomizu 1963]. Lifting distributions to tangent and jet bundles 17

Let us provide the finite-dimensional formulae for the Christoffel symbols for the two lifted affine connections. Let us first develop the notation for doing this. We suppose that we have a fixed affine connection ∇ on M defining and defined by a linear connection i HTM on πTTM : TTM → TM. We shall denote the Christoffel symbols for ∇ by γjk, i, j, k = 1, . . . , n. The Christoffel symbols for the affine connection ∇T on TM defined ∗ T α by the linear connection ΦM ((HTM) ) will be denoted Γβσ, α, β, σ = 1,..., 2n. The T Christoffel symbols for the affine connection ∇ on TM defined by the linear connection ∗ ∗ T α ΦM (coann((V TM) )) will be denoted Γβσ, α, β, σ = 1,..., 2n. One may then verify that for i, j, k = 1, . . . , n we have

∂γi Γi = γi , Γn+i = jk v`, jk jk jk ∂x` i Γj,k+n = 0, n+i i Γj,k+n = γjk, i Γj+n,k = 0, n+i i Γj+n,k = γjk, i Γj+n,k+n = 0, n+i Γj+n,k+n = 0, and ∂γi Γi = γi , Γn+i = `k v` − γi γmv`, jk jk jk ∂xj `m jk i Γj,k+n = 0, n+i Γj,k+n = 0, i Γj+n,k = 0, n+i i Γj+n,k = γjk, i Γj+n,k+n = 0, n+i Γj+n,k+n = 0.

5.6 Remarks: 1. If Z denotes the geodesic spray for ∇ and if ZT denotes the geodesic T T ∗ c spray for ∇ , then one readily verifies that Z = ΦM Z . 2. We note that the affine connection ∇T is the same as the “complete lift” of ∇ as defined in [Yano and Ishihara 1973]. As such, it is the unique affine connection on TM satisfying c T c (∇X Y ) = ∇Xc Y T for all vector fields X and Y on M. The author knows of no occurrence of ∇ in the existing literature, but it is entirely possible that one exists. •

6. When M is the total space of a bundle over R

We now let M be the total space of a locally trivial ﬁbre bundle π : M → R. We are not here interested in connections on this bundle as in Propositions 2.5 and 3.4. Rather we are interested in looking at when a distribution is “compatible” with the process of taking local sections c:[a, b] → M of π. Local sections are to be regarded as special classes of curves, so we are interested in distributions which admit this class of curves as integral curves. Also associated with sections of π are jets of sections. When a distribution is compatible with the process of taking sections, it also interacts nicely with the jet bundles. Now let us make this discussion precise. 18 A. D. Lewis

6.1. Jet bundle notation. We denote by J kM the bundle of k-jets of sections of π (we refer k l to [Saunders 1989] for jet bundles). For k > l we let τk,l : J M → J M be the projection 0 which forgets the higher order equivalence. We take J M = M and denote τk = τk,0. Denote by VM ⊂ TM the kernel of the projection T π and denote by νM : VM → M k k−1 the projection. Recall that the ﬁbre bundle τk,k−1 : J M → J M is an aﬃne bundle ∗ ∗ k−1 k−1 modelled on the pull-back vector bundle τk−1νM : τk−1VM → J M. If ξ ∈ J M, we k −1 denote by Jξ M = τk,k−1(ξ). The bundle π : M → R admits natural adapted charts since the base space R has a natural set of coordinates. Let us always assume we are using such an adapted chart, and we denote by (t, u) the coordinates in this chart where u ∈ U 0 and U 0 is an open subset of a Banach space E. We denote natural coordinates for J kM by k k−1 (t, u, u1, . . . , uk). Recall that J M is naturally a submanifold of T (J M). For k = 1, 2 these inclusions are given in natural coordinates by

(t, u, u1) 7→ ((t, u), (1, u1))

(t, u, u2, u2) 7→ ((t, u, u1), (1, u1, u2)).

k k−1 2 Let us denote the inclusion of J M in T (J M) by ιk. Note also that J M ⊂ TTM. In natural coordinates this inclusion is given by

(t, u, u1, u2) 7→ ((t, u, 1, u1), (1, u1, 0, u2)).

In this section we shall make these identiﬁcations of jet bundles as subsets of tangent bundles without warning.

6.2. Restricting lifted distributions to J 1M. Now let us suppose D to be a distribution on M. We wish to ascertain under what conditions the lifted distribution DT restricts to a distribution on J 1M ⊂ TM. We further wish to give some properties of the restriction 1 2 T of the lifted distribution in this case. Let us denote D1 = J M ∩ D and D2 = J M ∩ D . The following result tells the tale. 6.1 Proposition: If D is a distribution on M then the following are equivalent:

(i) dt 6∈ ann(Dx) for every x ∈ M; 1 (ii) Dx ∩ JxM 6= ∅ for every x ∈ M; (iii) for every x ∈ M there exists a local section c:[a, b] → M so that c(t) = x for some 0 t ∈ [a, b] and so that c (t) ∈ Dc(t) for every t ∈ [a, b]. Furthermore, any of the above statements implies each of the following: (iv) (DT |J 1M) ∩ T (J 1M) is a subbundle of DT |J 1M of codimension 1. 1 (v) D1 is an aﬃne subbundle of J M modelled on the subbundle D ∩ VM of VM; 2 (vi) D2|D1 is an aﬃne subbundle of J M|D1 modelled on the pull-back of D ∩ VM to D1 by τ1|D1.

Proof: Let us ﬁrst get our local representations down. We let (U, φ) be an adapted chart for π : M → R and we use the induced charts for TM and J 1M. Let us also adapt the local notation for D introduced in the proof of Proposition 2.2. Thus we write

D(t,u) = {((t, u), (B01(t, u) · f1,B11(t, u) · f1)) | f1 ∈ F1} Lifting distributions to tangent and jet bundles 19

where F1 is a Banach space and the map f1 7→ (B01(t, u) · f1,B11(t, u) · f1) ∈ R × E is injective with split image. Here B01 : U → L(F1; R) and B11 : U → L(F1; E). (i) ⇐⇒ (ii) We have

D1,(t,u) = {((t, u), (1,B11(t, u) · f1)) | B01(t, u) · f1 = 1}. (6.1)

This set is non-empty if and only if B01(t, u) 6= 0. However, (i) is clearly also equivalent to having B01(t, u) 6= 0. (ii) ⇐⇒ (iii) Suppose (ii) holds. Let f1 ∈ F1 be deﬁned by

(B01(t0, u0) · f1,B11(t0, u0) · f1) = (1, e)

0 for some (t0, u0) ∈ U and some e ∈ image(B11(t0, u0)). Let t 7→ (s(t), c(t)) ∈ R × U be the solution of the diﬀerential equation

s˙(t) = 1

c˙(t) = B11(t, c(t)) · f1 with initial condition (s(t0), c(t0)) = (t0, u0). Clearly the tangent vector field to the local section t 7→ (t, c(t)) lies in D. In other words, we have shown that (iii) holds. Now suppose (iii) holds and let c be a local section whose tangent vector field lies in D. If we locally write c as t 7→ (t, c(t)) then c0 has local representative t 7→ ((t, c(t)), (1, Dc(t) · 1)) 1 which means that (1, Dc(t) · 1) ∈ D(t,c(t)). This implies that Dx ∩ JxM is non-empty for each x ∈ M since such a section can be constructed so as to pass through any point x ∈ M. Now we move on to the second part of the proposition. We first need some notation. We have the local representation of T ι1 as

T ι1((t, u, u1), (τ, e1, e2)) = ((t, u, 1, u1), (τ, e1, 0, e2)). (6.2)

Using (2.5) we easily compute

T D(t,u,τ,e) = ((t, u, τ, e), (B01(t, u) · f11,B11(t, u) · f11,

D1B01(t, u) · (τ, f11) + D2B01(t, u) · (e, f11) + B01(t, u) · f12,

D1B11(t, u) · (τ, f11) + D2B11(t, u) · (e, f11) + B11(t, u) · f12)) | f11, f12 ∈ F1 . (6.3)

To restrict DT to J 1M set τ = 1. (i) =⇒ (iv) We ﬁrst claim that (i) implies that the subspace

{(f11, f12) ∈ F1 ×F1 | D1B01(t, u)·(1, f11)+D2B01(t, u)·(e, f11)+B01(t, u)·f12 = 0} (6.4) has codimension 1 in F1 × F1 for each (t, u, e) ∈ U × E. Indeed, since (i) is equivalent to B01(t, u) being non-zero for each (t, u) ∈ U, this claim follows. Now since the map

(f1, f2) 7→ D1B01(t, u) · (1, f11) + D2B01(t, u) · (e, f11) + B01(t, u) · f12 is continuous and R-valued, it has closed kernel. This shows that the fibres of (DT |J 1M) ∩ T (J 1M) are closed and of finite-codimension. This implies that these fibres split in the fibres of DT |J 1M and are of constant finite-codimension. This in turn implies (iv). 20 A. D. Lewis

(i) =⇒ (v) By the local form (6.1) of D1 we must show that the set

{(1, u1) ∈ R × E | u1 = B11(t, u) · f1 where B01(t, u) · f1 = 1} is an aﬃne space modelled on the subspace

{e ∈ E | e = B11(t, u) · f1 where B10(t, u) · f1 = 0} with the aﬃne action given by (e, (1, u1)) 7→ (1, u1 + e). This is a simple matter of verifying the axioms of an aﬃne space (see [Berger 1987]) using the fact that the map f1 7→ (B01(t, u)· f1,B11(t, u) · f1) is injective. (i) =⇒ (vi) Let (t, u, u1) ∈ U × E be the image of a point v ∈ D1. We have

D2,(t,u,u1) = ((t, u, 1, u1), (1, u1, 0,

D1B11(t, u) · (1, f1) + D2B11(t, u) · (u1, f1) + B11(t, u) · f2)) |

B01(t, u) · f1 = 1,B11(t, u) · f1 = u1,

D1B01(t, u) · (1, f1) + D2B01(t, u) · (u1, f1) + B01(t, u) · f2 = 0 .

The relations B01(t, u) · f1 = 1,B11(t, u) · f1 = u1 are satisﬁed since (t, u, u1) ∈ D1,(t,u). Note that these relations uniquely determine f1 ∈ F1. We shall write f1(t, u, u1) to denote this element of F1 for (t, u, u1) ∈ U × E. The relation

D1B01(t, u) · (1, f1) + D2B01(t, u) · (u1, f1) + B01(t, u) · f2 = 0 is satisﬁed since we are restricting to J 2M ⊂ T (J 1M). Our assertion will be proved if we can show that, for each (t, u, u1) ∈ U × E, the set

(1, u1, 0, u2) ∈ R × E × R × E | u2 = D1B11(t, u) · (1, f1(t, u, u1))+

D2B11(t, u) · (u1, f1(t, u, u1)) + B11(t, u) · f2) where

D1B01(t, u) · (1, f1(t, u, u1)) + D2B01(t, u) · (u1, f1(t, u, u1)) + B01(t, u) · f2 = 0 is an aﬃne space modelled on the subspace

{e ∈ E | e = B11(t, u) · f1 where B01(t, u) · f1 = 0} with the affine action given by (e, (1, u1, 0, u2)) 7→ (1, u1, 0, u2 + e). This verification of the axioms for an affine space follows from the fact that the map f1 7→ (B01(t, u)·f1,B11(t, u)·f1) is injective.

When a distribution satisfies the condition (i), we shall say it is compatible with π, and we denote the associated distribution (DT |J 1M) ∩ T (J 1M) on J 1M by j1D. If D is compatible with π then it is not difficult to see that we may inductively define jkD as a distribution on J kM by suitably restricting the distribution DT k on T kM to the submanifold J kM. Lifting distributions to tangent and jet bundles 21

In ﬁnite-dimensions, the situation is as follows. Let us denote by (t, xi) coordinates in an adapted chart, and by (t, xi, τ, vi) the associated natural coordinates for TM. Suppose we have a local basis ∂ ∂ X = X0 + Xi , a = 1, . . . , m a a ∂t a ∂xi 0 for D. Then dt 6∈ ann(D) means exactly that Xa should be non-zero for at least one a = 1, . . . , m. The restricted vector bundle DT |J 1M is then locally generated by

∂ ∂ Xv|J 1M = X0 + Xi , a = 1, . . . , m a a ∂τ a ∂vi 0 0 c 1 0 ∂ i ∂ ∂X ∂X j ∂ X |J M = X + X + b + b v + (6.5) b b ∂t b ∂xi ∂t ∂xj ∂τ ∂Xi ∂Xi ∂ b + b vj , b = 1, . . . , m. ∂t ∂xj ∂vi

A local basis for j1D consists of linear combinations of these 2m vector ﬁelds which satisfy the single linear equation (6.4). In ﬁnite-dimensions this equation reads

∂X0 ∂X0 X0Aa + a + a vi Ba = 0 a ∂t ∂xi which we may solve for Aa,Ba, a = 1, . . . , m, these then being coefficients for the vector fields (6.5) ensuring that they lie in j1D. Our assumption that dt 6∈ ann(D) implies that 0 this linear system has rank 1. Without loss of generality we may suppose that X1 (x) 6= 0 for all x in our coordinate neighbourhood on M. With this supposition one may verify that the 2m − 1 vector fields

6.3. Restricting lifted codistributions to J 1M. Now we turn to the case when Λ is a codistribution on the total space M of a locally trivial ﬁbre bundle π : M → R. The following ⊥ 1 result gives the analogue of Proposition 6.1 in this case. We denote Λ1 = coann(Λ) ∩ J M ⊥ T 2 and Λ2 = coann(Λ ) ∩ J M. 22 A. D. Lewis

6.2 Proposition: Let Λ be a codistribution on M. The following are equivalent:

(i) Λx ∧ dt 6= 0 (meaning βx ∧ dt 6= 0 for βx ∈ Λx \{0}) for every x ∈ M; 1 (ii) coann(Λx) ∩ JxM 6= ∅ for every x ∈ M; (iii) for every x ∈ M there exists a local section c:[a, b] → M so that c(t) = x for some 0 t ∈ [a, b] and so that c (t) Λc(t) = 0 for every t ∈ [a, b]. Furthermore, any of the above statements implies each of the following: ∗ T 1 (iv) Tv ι1|Λv is an injection for every v ∈ J M; ⊥ 1 (v) Λ1 is an aﬃne subbundle of J M modelled on the subbundle coann(Λ)∩VM of VM; ⊥ ⊥ 2 ⊥ (vi) Λ2 |Λ1 is an aﬃne subbundle of J M|Λ1 modelled on the pull-back of coann(Λ)∩VM ⊥ ⊥ to Λ1 by τ1|Λ1 .

Proof: We use the notation of Proposition 3.1. Thus we let (U, φ) be an adapted chart for M with natural chart (T U, T φ) for TM. We suppose φ is (R × E)-valued. Adapting the notation of Proposition 3.1 to our present situation of a locally trivial ﬁbre bundle, we have

2 2 2 ∗ Λ(t,u) = {((t, u), (Q10(t, u) · ν ,Q11(t, u) · ν )) | ν ∈ V2 }

∗ ∗ ∗ ∗ for some Banach space V2, and where Q10 : U → L(V2 ; R ) and Q11 : U → L(V2 ; E ). Note 2 2 2 that since Λ is a subbundle, the map ν 7→ (Q10(t, u) · ν ,Q11(t, u) · ν ) is injective with split image. The crux of the proof is the following lemma.

1 Lemma: In our local coordinate representation, (i) is equivalent to Q11(t, u) being injective for each (t, u) ∈ U.

Proof: In our local coordinate representation dt = (1, 0). Using the deﬁnition of the wedge product, (i) is equivalent to

2 (Q11(t, u) · ν ) · (τ2e1 − τ1e2) 6= 0

2 ∗ for every ν ∈ V2 \{0}, τ1, τ2 ∈ R, and e1, e2 ∈ E. This is clearly equivalent to (Q11(t, u) · 2 2 ∗ ν ) · e 6= 0 for every ν ∈ V2 \{0} and e ∈ E. But this is equivalent to the statement that 2 2 Q11(t, u) · ν = 0 if and only if ν = 0, i.e., that Q11(t, u) be injective. H

(i) ⇐⇒ (ii) This follows from Proposition 6.1(i) since (i) is equivalent to dt 6∈ Λx for every x ∈ M. (ii) ⇐⇒ (iii) This follows in exactly the same manner as did the similar step in the proof of Proposition 6.1. (i) =⇒ (iv) By (3.5) we have

T 21 21 Λ(t,u,τ,e) = ((t, u, τ, e), (D1Q10(t, u) · (·, ν , τ) + D1Q11(t, u) · (·, ν , τ)+ 22 21 21 22 Q10(t, u) · ν , D2Q10(t, u) · (·, ν , e) + D2Q11(t, u) · (·, ν , e) + Q11(t, u) · ν , 21 21 21 22 ∗ Q10(t, u) · ν ,Q11(t, u) · ν )) | ν , ν ∈ V2 .

∗ We also have the local representative of Tv ι1 as T ∗ ι (λ1, α1, λ2, α2) = ((t, u, u ), (λ1, α1, α2)). (t,u,1,u1) 1 1 Lifting distributions to tangent and jet bundles 23

Therefore, T ∗ ι |ΛT is injective if and only if the map (t,u,u1) 1

21 22 21 21 22 (ν , ν ) 7→ (D1Q10(t, u) · (·, ν , 1) + D1Q11(t, u) · (·, ν , 1) + Q10(t, u) · ν , 21 21 22 21 D2Q10(t, u) · (·, ν , u1) + D2Q11(t, u) · (·, ν , u1) + Q11(t, u) · ν ,Q11(t, u) · ν )

2 2 2 is injective. By Lemma1 and by the fact that the map ν 7→ (Q01(t, u) · ν ,Q11(t, u) · ν ) is injective, the result follows. (i) =⇒ (v) We have

⊥ ∗ ∗ Λ1 = {((t, u), (1, u1)) | Q10(t, u) · 1 + Q11(t, u) · u1 = 0}. (6.6) Our assertion will follow if we can show that for ﬁxed (t, u) the set

∗ ∗ {(1, u1) ∈ R × E | Q11(t, u) · u1 + Q10(t, u) · 1 = 0} is an aﬃne space modelled on the set

∗ {e ∈ E | Q11(t, u) · e = 0} with the affine action given by (e, (1, u1)) 7→ (1, u1 + e). This is a simple matter of verifying the axioms for an affine space. ⊥ (i) =⇒ (vi) Using the definition (6.6) of Λ1 , we may see that we must verify that for fixed (t, u, u1) the set

∗ ∗ (1, u1, 0, u2) ∈ R × E × R × E | Q11(t, u) · u2 + D2Q10(t, u) · (u1, 1)+ ∗ ∗ ∗ D2Q11(t, u) · (u1, u1) + D1Q10(t, u) · (1, 1) + D1Q11(t, u) · (1, u1) = 0 is an aﬃne space modelled on the subspace

∗ {e ∈ E | Q11(t, u) · e = 0} with the affine action given by (e, (1, u1, 0, u2)) 7→ (1, u1, 0, u2 + e). Again, this is a simple matter of verifying the affine space axioms. If a codistribution Λ satisfies (i) we shall say it is compatible with π. The codistribution 1 ∗ T 1 on J M defined by v 7→ Tv ι1(Λv ) we will denote as j Λ. As with distributions compatible with π, if Λ is compatible with π then we may lift Λ to a codistribution jkΛ on J kM by restricting ΛT k to J kM ⊂ T kM. Let us see what this looks like in finite-dimensions. We write a local basis for Λ as

a a i a β = βi dx + β0 dt, a = 1, . . . , m.

a The condition (i) is equivalent in this case to the matrix with components βi having maximal rank. A local basis for j1Λ is then

a a i β0 dt + βi dx , a = 1, . . . , m ∂βb ∂βb ∂βb ∂βb 0 + i vi dt + 0 + i vi dxj + βbdvi, b = 1, . . . , m. ∂t ∂t ∂xj ∂xj i These formulas are also produced by de Le´on,Marrero, and Mart´ınde Diego [1997]. 24 A. D. Lewis

6.4. When the codistribution and distribution annihilate one another. Of course, the results of Section4 still hold when D and Λ = ann(D) are defined on the total space of the fibration π : M → R. However, the jet bundle structure comes into play to provide extra and interesting structure, even when (ann(D))T 6= ann(DT ). The gist of the following result is that even though one cannot generally expect ΛT = ann(DT ), the jet bundle constructions of the previous two sections are unable to distinguish between when one uses distributions or when one uses codistributions. That is to say, ΛT and ann(DT ) differ on that part of T (J 1M) which does not intersect J 2M. 6.3 Proposition: Let D be a distribution on the total space M of a locally trivial fibre bundle π : M → R and let Λ = ann(D) be its annihilating codistribution. The following statements hold: (i) D is compatible with π if and only if Λ is compatible with π; ⊥ (ii) D1 = Λ1 ; ⊥ ⊥ (iii) D2|D1 = Λ2 |Λ1 .

Proof: (i) This is a consequence of the fact that condition (i) of Proposition 6.1 and condition (i) of Proposition 6.2 are equivalent if Λ = ann(D). (ii) This is obvious since Λ = ann(D). (iii) We resume with the notation of the proof of Propositions 6.1 and 6.2. We also use some notation from the proof of Proposition 2.2 which we did not use in Proposition 6.1: we denote by

(f1, f2) 7→ (B01(t, u) · f1 + B02(t, u) · f2,B11(t, u) · f1 + B12(t, u) · f2) the full parameterisation of the tangent space to M at (t, u) in a vector bundle chart adapted to D. Thus B0i : U → L(Fi; R) and B1i : U → L(Fi; E) for i = 1, 2. We denote the inverse of this map by

(τ, e) 7→ (A10(t, u) · τ + A11(t, u) · e, A20 · τ + A21(t, u) · e) where Ai0 : U → L(R; Fi) and Ai1 : U → L(E; Fi), i = 1, 2. Thus

D(t,u) = {((t, u), (B01(t, u) · f1,B11(t, u) · f1)) | f1 ∈ F1} as before. Since we are assuming Λ = ann(D) we may take

∗ ∗ ∗ ∗ ((t, u), (λ, α)) 7→ ((t, u), (B01(t, u) · λ + B11(t, u) · α, B02(t, u) · λ + B11(t, u) · α)) as a vector bundle chart for T ∗M adapted to Λ. Thus

∗ 2 ∗ 2 2 ∗ Λ(t,u) = {((t, u), (A20(t, u) · ν ,A21(t, u) · ν )) | ν ∈ F2 }.

We must then have

∗ 2 ∗ 2 (A20(t, u) · ν ) · (B01(t, u) · f1) + (A21(t, u) · ν ) · (B11(t, u) · f1) = 0 Lifting distributions to tangent and jet bundles 25

2 ∗ for all f1 ∈ F1 and ν ∈ F2 . Diﬀerentiating this in the direction of (τ, e) ∈ R × E gives

∗ 2 ∗ 2 (D1A20(t, u) · (τ, ν )) · (B01(t, u) · f1) + (D2A20(t, u) · (e, ν )) · (B01(t, u) · f1)+ ∗ 2 ∗ 2 (A20(t, u) · ν ) · (D1B01(t, u) · (τ, f1)) + (A20(t, u) · ν ) · (D2B01(t, u) · (e, f1))+ ∗ 2 ∗ 2 (D1A21(t, u) · (τ, ν )) · (B11(t, u) · f1) + (D2A21(t, u) · (e, ν )) · (B11(t, u) · f1)+ ∗ 2 ∗ 2 (A21(t, u) · ν ) · (D1B11(t, u) · (τ, f1)) + (A21(t, u) · ν ) · (D2B11(t, u) · (e, f1)) = 0. (6.7)

1 Now let (t, u, u1) lie in the intersection of D with J M. Therefore, for some uniquely determined f1 ∈ F1 we have

(B01(t, u) · f1,B11(t, u) · f1) = (1, u1). (6.8)

We denote this value of f1 by f1(t, u, u1). Since Λ = ann(D) we also have

∗ 2 ∗ 2 (A20(t, u) · ν ) · u1 + (A21(t, u) · ν ) · 1 = 0 (6.9)

2 ∗ for ν ∈ F2 . We have

DT = ((t, u, 1, u ), (1, u , D B (t, u) · (1, f (t, u, u ))+ (t,u,1,u1) 1 1 1 01 1 1

D2B01(t, u) · (u1, f1(t, u, u1)) + B01(t, u) · f2, D1B11(t, u) · (1, f1(t, u, u1))+

D2B11(t, u) · (u1, f1(t, u, u1)) + B11(t, u) · f2)) | f1 ∈ F1 . (6.10) and

ΛT = ((t, u, 1, u ), (D A∗ (t, u) · (·, ν21, 1) + D A∗ (t, u) · (·, ν21, 1)+ (t,u,1,u1) 1 1 20 1 21 ∗ 22 ∗ 21 ∗ 21 ∗ 22 A20(t, u) · ν , D2A20(t, u) · (·, ν , u1) + D2A21(t, u) · (·, ν , u1) + A21(t, u) · ν , ∗ 21 ∗ 21 21 22 ∗ A20(t, u) · ν ,A21(t, u) · ν )) | ν , ν ∈ F2 . (6.11)

Combining (6.7)–(6.11) with a tedious but straightforward calculation gives DT (t,u,1,u1) ΛT = 0 which proves the result, given that (t, u, u ) ∈ D . (t,u,1,u1) 1 1,(t,u)

6.4 Remark: As stated in the introduction, our constructions involving the lifting of distributions are applicable in the investigation of mechanical systems with aﬃne constraints [de Le´on,Marrero, and Mart´ınde Diego 1997]. Within this context, one may interpret the previous result as stating that our two ways of lifting a distribution are equally valid when applied to constrained mechanical systems. •

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