On the Construction of the Linearization of a Nonlinear Connection

Home , Jet bundle, Pullback bundle

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Introduction 1 π ∗ E ´ INEZ ymaso oain derivative covariant a of means by 7 sacknowledged. is 17R π 6-21PadPGC2018- and 166-C2-1-P : E e tnadopera- standard her neadusefulness and ance → c otecoordi- the to ect ytm defined systems l connection fteresultant the of pproxima- h enn of meaning the ta equations ntial oe,intrinsic roper, rvnt be to proven s hyaethe are they s M ais[,25], [5, hanics ytmi lin- is system e geometry, ler to r the are ction .Ti map This s. ,weethe where y, ftehor- the of n h resulting the ierap- linear t covariant e produces e edefined we e several ve, eometrical onnection eshown be a re- was t - 2 E. MARTÍNEZ non linear connection is actually homogeneous and hence not smooth on the zero section. From the horizontal lift we will obtain the expression of the associated covariant derivative, obtaining the formula which served as the definition of the linearized connection in [18]. The above interpretation allows to obtain the horizontal lift of a vector field as the fibre derivative of such a vector field. As a consequence, the flow of the horizontal lift of a vector field is but the fibre derivative of the flow of the given vector field, and hence it provides a detailed description of the parallel transport system associated to the linearized connection, from where we will derive some explicit new formulas for its curvature. A connection on a fiber bundle can be understood as a section of the first jet bundle of sections of that bundle. It is well known [12] that by taking the fibre derivative (or the vertical derivative) of such section one obtains a linear connection which however does not coincide with the linearization defined in this paper, as it is defined on a different bundle. It will be shown that composing with a canonical map we obtain a section of the first jet bundle of the pullback bundle which coincides with the linearization. An equivalent version on the vertical bundle will also be provided. The linearization of a nonlinear connection is a natural first order prolongation of the original connection. That is, their construction is invariant under fibered diffeomorphism, providing hence a geometric object, and uses only the first order derivatives of the initial connection. In the finite dimensional case, all natural first order prolongations of a connection were classified in [12] obtaining a 1-parameter family of connections. The linearized connection is the only member of that family which is linear. We will provide a direct proof of this fact, which is valid for connections on Banach bundles. Moreover, it will be shown that the linearization can be also characterized as the only semibasic connection in the family, a property which is easier to work with in practical computations. The paper is organized as follows. In Section 2 we will provide the necessary preliminary results. In Section 3 it will be shown that by linearizing the horizontal lifting map in an adequate sense we obtain a linear connection on the pullback bundle. In Section 4 we will show that all natural first order prolongations of a connection fit into a 1-parameter family of connections on the pullback bundle, and we characterize the linearization as the only element in the family which is linear. In Section 6 we will show the relation between the horizontal lift (with respect to the linearized connection) of a vector field and the fibre derivative of such vector field, and we will clarify the meaning of the parallel transport and of the curvature for the linearized connection. Finally, in Section 7 we will reconsider the problem from the perspective of jet bundle theory, and we will construct the section associated to the linearized connection from the section associated to the original non linear one. Throughout this paper a manifold will be understood as a smooth Banach manifold, and a fiber bundle will mean a smooth Banach fiber bundle (see [1, 13]). All objects will be considered in the C∞-smooth category.

2. Preliminaries

The tangent bundle to a manifold M will be denoted τM : T M → M, and the C∞(M)-module of vector ﬁelds on M will be denoted X(M). For a ﬁber bundle τ : P → M, the set of sections of τ will be denoted either by Sec(τ) or by Sec(P ) whenever the projection is clear. ON THE LINEARIZATIONOF A NON LINEAR CONNECTION 3

Given a map f : N → M and a fiber bundle τ : P → M, the pullback bundle f ∗τ : f ∗P → N has total space f ∗P = { (n, p) ∈ N × P | τ(p)= f(n) } and projection f ∗τ(n, p)= n. A section of f ∗τ : f ∗P → N is equivalent to a section of P along f, that is, a map σ : N → P such that τ ◦ σ = f. The set of sections along f will be denoted by Secf (τ) or Secf (P ). For the details see [24, 7]. We will mainly be concerned with the case of a vector bundle π : E → M. The pullback f ∗π : f ∗E → N is also a vector bundle. More concretely, we are interested in the case where the map f is just the projection π. A section σ ∈ Sec(π∗E) is said to be basic if there exists a section α ∈ Sec(E) such that σ = α ◦ π. We will frequently identify both sections, i.e. α indicates both the section of E and the section of π∗E, and the context will make clear the meaning. However, in the last section we will need to be more precise. V The vertical subbundle τE ≡ τE : Ver π = Ker(T π) → E is canonically ∗Ver π ∗ isomorphic to the pullback bundle π π : π E → E. The isomorphism is the vertical V ∗ V d lifting map ξπ : π E → Ver(π) ⊂ T E, defined by ξπ(a, b) = ds (a + sb)|s=0 for every ∗ (a, b) ∈ π E. The vertical projection is the map νπ : Ver π → E given by νπ = V −1 V ∗ ∗ pr2 ◦(ξπ) , that is νπ ξπ(a, b) = b for (a, b) ∈ π E. For a section σ ∈ Sec(π E) the V X V V vertical lift of σ is the vector field σ ∈ (E) given by σ = ξπ ◦ σ. For a section σ of E along π we write σV for the vertical lift of the section of π∗E associated to σ. V Clearly we have νπ ◦ σ = σ for σ ∈ Secπ(E) ∗ ∗ ∗ The tangent bundle τπ∗E : T (π E) → π E to the manifold π E can be canonically identified with the pullback bundle (T π)∗(T E). The identification is given by · (ω1,ω2) (0) ≃ ω˙ 1(0), ω˙ 2(0) for ω1,ω2 curves in E such that π ◦ ω1 = π ◦ ω2. Along ∗ ∗ this paper we will use such identification T (π E) ≃ (T π) (T E), and therefore a ∗ tangent vector to π E at the point (a, b) will be considered as a pair (w1,w2) ∈ TaE × TbE such that T π(w1)= T π(w2).

Connections. For a vector bundle π : E → M there is a short exact sequence of vector bundles over E

ξV p 0 −→ π∗E −→π T E −→E π∗T M −→ 0,

∗ where pE : T E → π (T M) is the projection pE(w)=(τE(w), T π(w)). This sequence is known as the fundamental sequence of π : E → M. An Ehresmann connection on the vector bundle E is a (right) splitting of the H ∗ H fundamental sequence, that is, a map ξ : π T M → T E such that pE ◦ξ = idπ∗(TM). The map ξH is said to be the horizontal lifting. Equivalently, a connection is given by the associated left splitting of that sequence, that is, the map ϑ: T E → π∗E V H ∗ such that ϑ ◦ ξπ = idπ E and ϑ ◦ ξ = 0. Closely related to the map ϑ is the the connector κ = pr2 ◦ϑ: T E → E, also known as the Dombrowski connection map. See [24] for the general theory of connections on vector bundles. A connection decomposes T E as a direct sum T E = Hor ⊕ Ver E, where Hor = Im(ξH) = Ker(ϑ) is said to be the horizontal subbundle. The projectors over the H V horizontal and vertical subbundles are given by PH = ξ ◦ pE and PV = ξπ ◦ ϑ. For a section X ∈ Sec(π∗T M) the horizontal lift of X is the vector field X H ∈ X(E) given H H H by X = ξ ◦ X. For a vector field X ∈ Secπ(T M) along π the symbol X denotes the horizontal lift of the associated section of π∗(T M). The curvature of the non linear connection is the E-valued 2-form R: π∗(T M ∧ T M) → π∗E defined by

R(X,Y )= ϑ([X H,Y H]),X,Y ∈ Sec(π∗T M). 4 E. MART´INEZ

A connection ξH on the pullback bundle π∗π : π∗E → E is therefore given by a map ξH : (π∗π)∗(T E) → T (π∗E) of the form ξH (a, b),w = w, B(a, b)w for B(a, b): TaE → TbE a linear map such that Tbπ = B(a, b) ◦ Taπ. A connection ξH on π∗π : π∗E → E is said to be basic if it is the pullback of a connection ξH on π : E → M, i.e. it is of the form ξH((a, b),w)=(w,ξH(b, T π(w))) ∗ H for all (a, b) ∈ π E and w ∈ TaE. The connection ξ is said to be semibasic if ∗ for all (a, b) ∈ π E the horizontal lift of a vertical vector w ∈ Vera(π) is of the H form ξ ((a, b),w)=(w, 0b). Obviously a basic connection is semibasic but the converse does not hold. In terms of the connector, the connection is semibasic if κ(w, 0b)=0π(b) for every vertical w ∈ Vera(π). Double vector bundle structure. For a vector bundle π : E → M the manifold T E has two different vector bundle structures. On one hand we have the standard tangent bundle τE : T E → E with operations denoted by + and · (the dot usually omitted in the notation). On the other, we also have a vector bundle structure T π : T E → T M where the operations are the differential of the operations in π : E → M and which will be denoted +T and ·T. They can be easily defined in terms of vectors tangent to curves by dα d(λα) λ ·T (0) = (0) dt dt and dα dβ d(α + β) (0) +T (0) = (0), dt dt dt where the curves α and β satisfy π ◦ α = π ◦ β. These operations are compatible, in the sense that they define a structure of double vector bundle on T E over E and T M. We will list in the next paragraphs a few properties, which are consequences of such a double structure, and that will be needed later on. For more details on double vector bundles see [14, 11]. The two different operations of addition are compatible in the sense that they satisfy the interchange law

(w1 + w2)+T (w3 + w4)=(w1 +T w3)+(w2 +T w4), for w1,w2,w3,w4 ∈ T E such that τE(w1)= τE(w2), τE(w3)= τE(w4) and T π(w1)= T π(w3), T π(w2)= T π(w4). Moreover, we have the following properties, which apply when the operations on the left hand side are deﬁned,

τE(v1 + v2)= τE(v1)= τE(v2) T π(v1 + v2)= T π(v1)+ T π(v2)

τE(λv)= τE(v) T π(λv)= λT π(v)

τE(v1 +T v2)= τE(v1)+ τE(v2) T π(v1 +T v2)= T π(v1)= T π(v2)

τE(λ ·T v)= λτE(v) T π(λ ·T v)= T π(v).

A connection ξH : π∗T M → T E on the vector bundle π : E → M is said to be linear if h(a, v) is linear in the argument a for every ﬁxed v ∈ T M, that is, if it satisﬁes H ′ H H ′ ξ (a + a , v)= ξ (a, v)+T ξ (a , v) H H ξ (λa, v)= λ ·T (ξ (a, v)), for every λ ∈ R, every (a, v) ∈ π∗T M and a′ ∈ E with π(a) = π(a′). A linear connection can be equivalently described by means of a covariant derivative operator D : X(M) × Sec(E) → Sec(E). The relation between the covariant derivative D ON THE LINEARIZATIONOF A NON LINEAR CONNECTION 5

H H and the connection ξ is given by Dvσ = νπ T σ(v) − ξ (σ(m), v) = κ(T σ(v)) for v ∈ TmM and σ ∈ Sec(E).

Both τE : T E → E and T π : T E → T M being vector bundles have corresponding associated vertical lifting maps to T (T E). On one hand we have the map V ∗ ξτE : τE(T E) → Ver(τE) ⊂ T T E given by

V d ξτE (v,w)= v + sw , where τE(v)= τE(w), ds s=0 V ∗ and on the other ξTπ : (T π) (T E) → Ver( T π) ⊂ T E given by

V d ξTπ(v,w)= v +T s ·T w , where T π(v)= T π(w). ds s=0 Correspondingly, we have two vertical projection maps, ντE : Ver(τE) → T E de- V V ﬁned by ντE (ξτE (v,w)) = w, and νTπ : Ver(T π) → T E deﬁned by νTπ(ξTπ(v,w)) = w. For the constructions in this paper the most relevant one is νTπ, which has the following properties,

νTπ(v1 +T v2)= νTπ(v1)+ νTπ(v2)

νTπ(λ ·T v)= λνTπ(v)

νTπ(λv)= λ ·T νTπ(v). Some auxiliary results. We will need the following auxiliary properties which can be found scattered in the literature. Proposition 2.1: The following properties hold.

(1) τE ◦ νTπ = νπ ◦ T τE Ver(Tπ).

(2) For v,w ∈ T E such that T π(v)= T π(w), V V T τE ξTπ(v,w) = ξτE τE(v), τE(w) . (3) For v,w ∈ Ver(π) such that τE(v)= τE(w), V V T νπ ξτE (v,w) = ξπ νπ(v), νπ(w) . Proof. (2) If v,w ∈ Ver(π) are such that T π(v)= T π(w) then

V d T τE ξTπ(v,w) = T τE (v +T s ·T w) ds s=0 d = τE v +T s ·T w) ds s=0 d = τE(v)+ sτE(w) ds s=0 V = ξτE τE(v), τE(w) . V (1) We write V ∈ Verv(T π) in the form V = ξTπ(v,w) for a unique w ∈ T E such that T π(v)= T π(w). Then, using (2) we have V V νπ ◦ T τE(V )= νπ ◦ T τE(ξTπ(v,w)) = νπ ξτE τE(v), τE(w) = τE(w) and on the other hand d τE ◦ νTπ(V )= τE ◦ νTπ (v +T s ·T w) = τE(w). ds s=0 V V (3) If v,w ∈ Vera(π) we can write v = ξπ(a, b) and w = ξπ(a, c) for some b, c ∈ Eπ(a). Then

V d V V d V ξτE (v,w)= [ξπ(a, b)+ sξπ(a, c)] = ξπ(a, b + sc) , ds s=0 ds s=0

6 E. MART´INEZ and hence

V d V T νπ ξτE (v,w) = T νπ ξπ(a, b + sc) ds s=0 d V = νπ ξE(a, b + sc) ds s=0 d = (b + sc) ds s=0 V = ξπ(b, c).

This proves the third property, as νπ(v)= b and νπ(w)= c.

On the double tangent bundle to a manifold P we have a canonical involution χTP : T (TP ) → T (TP ) which intertwines the two vector bundle structures τTP : TTP → TP and T τP : TTP → TP , i.e. χTP ◦ χTP = idTTP , τTP ◦ χTP = T τP . It is easily deﬁned in terms of 1-parameter families of curves θ : R2 → P by d d d d χTP θ(s, t) = θ(s, t) . dsdt t=0 s=0 dtds s=0 t=0

It satisﬁes

χTP (v1 + v2)= χTP (v1)+T χTP (v2) χTP (λ ·T v)= λχTP (v)

χTP (v1 +T v2)= χTP (v1)+ χTP (v2) χTP (λv)= λ ·T χTP (v).

In the concrete case P = E we also have the following properties that will be used later on. Proposition 2.2: Let π : E → M be a ﬁbre bundle. The canonical involution χTE : T T E → T T E satisﬁes

V V (1) ξTπ = χTE ◦ Tξπ. (2) νTπ = T νπ ◦ χTE|Ver(Tπ). (3) χTE restricts to a diﬀeomorphism χTE : Ver(T π) → T Ver(π). Proof. (1) Given (v,w) ∈ (T π)∗T E ≃ T (π∗E) we consider a pair of curves α(t), β(t) in E such that π ◦ α = π ◦ β and v =α ˙ (0), w = β˙(0). Then

V d V χTE ◦ Tξπ(v,w)= χTE ξπ α(t), β(t) dt t=0 d d = χTE α(t)+ sβ( t) dt ds s=0 t=0 d d = α(t)+ sβ(t) ds dt t=0 s=0 d = v +T s ·T w ds s=0 V = ξTπ(v,w).

V −1 V −1 (2) Taking inverses in the above equation we get (ξTπ) = (Tξπ) ◦ χTE, and taking the second component we obtain νTπ = T νπ ◦ χTE|Ver(Tπ). V V (3) The third property follows from the ﬁrst and the fact that ξTπ and ξπ are isomorphisms. ON THE LINEARIZATIONOF A NON LINEAR CONNECTION 7

3. Linearization of a non linear connection on a vector bundle Consider a nonlinear connection Hor on a vector bundle π : E → M given by a H ∗ horizontal lifting map ξ : π T M → T E. Fix a point a ∈ E and a vector w ∈ TaE tangent to E at the point a. Setting m = π(a)andw ¯ = T π(w) ∈ TmM we consider H −1 the map c ∈ Em 7→ ξ (c, w¯) ∈ T E. The image of this map is in the fibre (T π) (¯w) H because T π(ξ (c, w¯))=w ¯ for all c ∈ Em. Therefore we have a map between two vector spaces −1 −1 H ψw¯ : π (m) → (T π) (¯w), c 7→ ξ (c, w¯). H The connection ξ is linear if and only if the map ψw¯ is linear for every fixedw ¯. H When ξ is not linear, the differential of the map ψw¯ at the point a, the best linear approximation to ψw¯ at a, is a linear map b 7→ Dψv(a)b between the same vector −1 spaces. The vector Dψw¯(a)b is thus an element of (T π) (¯w) ⊂ T E. One has to be careful with the differential Dψw¯(a)b because the operations (sum and product by scalars) involved in this limit are the ones in the vector space (T π)−1(v), i.e. those of the vector bundle T π : T E → T M. In other words the precise meaning of the differential Dψw¯(a)b is 1 Dψw¯(a)b = lim ·T [ψw¯ (a + sb) −T ψw¯(a)] s→0 s

1 H H = lim ·T [ξ (a + sb, w¯) −T ξ (a, w¯)]. s→0 s

The vector Dψw¯(a)b is tangent to E at the point b. In this way, for (a, b) ∈ ∗ π E we have defined a linear map B(a, b): w ∈ TaE 7→ Dψw¯(a)b ∈ TbE which satisfies Tbπ(B(a, b)w)= Taπ(w), and hence it defines a horizontal lift (a, b),w 7→ w, B(a, b)w on the pullback bundle π∗E → E, which will be shown to be a linear connection on such bundle. To avoid as much as possible the explicit use of limits, instead of working with the differential in the above vector spaces we will make use of the canonical identification of a vector tangent to the fibre with a vertical vector. That is, we will consider the vector tangent to the curve s 7→ ξH(a+sb, w¯), which is a T π-vertical vector, and then we take the element in the fibre corresponding to it. In this way, we can equivalently define d H B(a, b)w = νTπ ξ (a + sb, T π(w)) , ds s=0 which is the expression to be used in what follows. Theorem 3.1: Let Hor be an Ehresmann connection on a vector bundle π : E → M H ∗ with horizontal lift ξ . For (a, b) ∈ π E and w ∈ TaE let B(a, b)w ∈ T E be the vector d H B(a, b)w = νTπ ξ (a + sb, T π(w)) . ds s=0 H¯ ∗ ∗ H¯ Then the map ξ : π E ×E T E → T (π E) given by ξ (a, b),w = w, B(a, b)w is the horizontal lift of a linear connection Hor on the pullback vector bundle π∗π : π∗E → E. ∗ Proof. Given (a, b) ∈ π E and w ∈ TaE we consider the curve α: R → T E given by α(s)= ξH a + sb, T π(w) . In terms of this curve, the vector B(a, b)w is B(a, b)w = νTπ(α ˙ (0)). The curve α depends on a, b, and w. At every step in the proof, when needed, we will indicate with a subscript the dependence that is relevant for the argumentation. 8 E. MARTÍNEZ

(1) The map B is well deﬁned and B(a, b)w ∈ TbE. Indeed, from (T π ◦ α)(s)= T π(w), constant, we get that T T π(α ˙ (0))=0. Thusα ˙ (0) ∈ Ver(T π) and we can apply νTπ to it. On the other hand, using Proposition 2.1 (1) we get

τE B(a, b)w = τE ◦ νTπ(α ˙ (0))

= νπ ◦ T τE(α ˙ (0)) d = νπ (τE ◦ α)(0) ds d = νπ (a + sb) ds s=0

= b.

(2) B(a, b)w is linear in w. Indeed, we have that ξH(a, λv) = λξH(a, v), from H where αw(s)= ξ (a+sb, T π(w)) satisﬁes αλw(t)= λαw(t) and henceα ˙ λw(0) = λ ·T α˙ w(0). Therefore

B(a, b)(λw)= νTπ(α ˙ λw(0))

= νTπ(λ ·T α˙ w(0))

= λνTπ(α ˙ w(0)) = λB(a, b)w.

Thus B(a, b) is a linear map from TaE to TbE. ∗ (3) w, B(a, b)w ∈ T(a,b)π E. Indeed, T π(B(a, b)w)= T π ◦ νTπ(α ˙ (0))

= T π ◦ τTE(α ˙ (0)) = T π(α(0)) = T π(w).

By (1), (2) and (3) we get that (a, b),w 7→ w, B(a, b)w is the horizontal lift of a connection Hor on π∗π : π∗E →E. (4) The connection Hor is linear. We have to prove linearity of B(a, b)w in b. In view of smoothness of B we just need to prove homogeneity, that is, H B(a, λb)w = λ ·T B(a, b)w for all λ ∈ R. The curve αb(s)= ξ (a + sb, T π(w)) satisﬁes αλb(s)= αb(λs), from whereα ˙ λb(0) = λα˙ b(0). Hence

B(a, λb)w = νTπ(α ˙ λb(0))

= νTπ(λα˙ b(0))

= λ ·T νTπ(α ˙ b(0))

= λ ·T B(a, b)w.

This ends the proof

The connection Hor is said to be the linearization of the non linear connection Hor. ∗ ∗ It is a semibasic linear connection on π E. Indeed, if (a, b) ∈ π E and w ∈ Vera(π) H¯ then B(a, b)w =0b, so that ξ ((a, b),w)=(w, 0b). ON THE LINEARIZATIONOF A NON LINEAR CONNECTION 9

If Hor is a linear connection, then the linearization Hor is basic and equal to the ∗ pullback of Hor. Indeed, if (a, b) ∈ π E and w ∈ TaE then

d H B(a, b)w = νTπ ξ (a + sb, T π(w)) ds s=0

d H H = νTπ ξ a, T π(w) +T s ·T ξ a, T π(w) ds s=0 V H H = νTπ ξTπ(ξ (a, T π(w)),ξ (b, T π(w))) = ξH(b, T π(w)). Therefore ξH¯((a, b),w)=(w,ξH(b, T π(w))) and hence Hor is the pullback of Hor. Coordinate expressions. Asume that E is finite dimensional. Consider local coordi- i nates (x ) defined on an open subset U ⊂ M and a local basis {eA} of local sections of E defined on U. If a point m ∈ U has coordinates (xi) and the components A A of a vector a ∈ Em in the basis {eA(m)} are (y ) (i.e. a = y eA(m)) then the coordinates of a are (xi,yA). Let us find the expression of the map B in local coordinates as above. If the point a ∈ E has coordinates (x, y), the point b ∈ E has coordinates (x, z), and i A the the tangent vector w ∈ TaE has components (w ,w ) in the coordinate basis {∂/∂xi,∂/∂yA}, then the coordinate expression of the curve α(s)= ξH(a+sv,Tπ(w)) is i A A i A i α(s)= x ,y + sz , w , −Γi (x, y + sz)w . Taking the derivative at s =0 we get i A i A i A A B i α˙ (0) = x ,y ,w , −Γi (x, y)w , 0, z , 0, −ΓiB(x, y)z w A A B where ΓiB(x, y)= ∂Γi /∂y (x, y). Finally applying νTπ we get i A i A B i B(a, b)w = x , z ,w , −ΓiB(x, y)z w . This expression shows clearly that B(a, b)w is linear both in b and w. An equivalent expression is

i ∂ A ∂ i ∂ A B ∂ B(a, b) w i + w A = w i − ΓiB(x, y)z A , ∂x (x,y) ∂y (x,y) ∂x (x,z) ∂y (x,z) and thus the horizontal lift is given by

H¯ i ∂ A ∂ ξ (a, b), w i + w A = ∂x (x,y) ∂y (x,y)

i ∂ A ∂ i ∂ A B i ∂ = w i + w A , w i − ΓiB(x, y)z w A . ∂x (x,y) ∂y (x,y) ∂x (x,z) ∂y (x,z)

A local basis of the horizontal distribution is ¯ ∂ ∂ A B ∂ Hi = i , i − ΓiB(x, y)z A , ∂x (x,y) ∂x (x,z) ∂y (x,z)

¯ ∂ HA = A , 0(x,z) . ∂y (x,y)

¯ A Together with VA = (0(x,y),∂/∂z (x,z)) they form a local basis of vector ﬁelds on the manifold π∗E.

In connection theory it is customary to indicate locally a connection as the Pfaff system given by the annihilator of the horizontal distribution. For the original A A i connection Hor such a Pfaff system is dy +Γi (x, y)dx = 0. For the linearized 10 E. MARTÍNEZ

A A B i connection Hor it is given by dz +ΓiB(x, y)z dx = 0. Thus, symbolically, the linearization procedure consist in A A i A A B i Hor : dy +Γi (x, y)dx =0 Hor : dz +ΓiB(x, y)z dx =0. n n ∂ΓA where we recall that ΓA = i . iB ∂yB

Restricted domain. In many situations the initial non linear connection is not defined (or is not smooth) on the whole vector bundle E. A typical example occurs in Finsler geometry where E = T M and the connection is defined by a Finsler metric. The most notable Finsler connections are homogeneous (generally non linear) and are smooth on the slit tangent bundle ˚τM : T˚ M → M, i.e. on the tangent bundle to the base manifold without the zero section. In this respect, at step (4) in the proof of Theorem 3.1, we have assumed smoothness of B(a, b) at b = 0 and hence linearity followed from homogeneity. In the case of a restricted domain (not including the zero section) Theorem 3.1 only proves homogeneity of the ‘linearized’ connection Hor. However, as it is clear from the coordinate expressions above, the connection coefficients are indeed linear in the variable z. We will next prove that the connection Hor is in fact linear even in this more general case. Theorem 3.2: Let C ⊂ E be a submanifold of E such that π−1(m) ∩ C is a non-empty open subset of Em for every m ∈ M, and denote by πC : C → M the restriction of π to C. Given a nonlinear connection Hor on the bundle π : E → M, smooth in the submanifold C, the map B defined as in Theorem 3.1 provides a linear ∗ ∗ ∗ connection on the pullback bundle π E with domain πC E = { (a, b) ∈ π E | a ∈ C }. Proof. It can be easily checked that all the steps in the proof of Theorem 3.1 remains valid if the domain of Hor is the submanifold C of E. To prove linearity we just need to show that the family B satisfies the property B(a, b1 + b2)w = B(a, b1)w +T B(a, b2)w. H Consider the curve αb(s)= ξ (a + sb, T π(w)) as above, and the map A(s1,s2)= H ξ (a + s1b1 + s2b2, T π(w)). Then A(s,s)= αb1+b2 (s), A(s, 0) = αb1 (s) and A(0,s)=

αb2 (s), so that d B(a, b1 + b2)w = νTπ αb1+b2 (s) ds s=0

d = νTπ A(s,s) ds s=0

∂A ∂A = νTπ (0, 0)+ (0, 0) ∂s1 ∂s2 d d = νTπ αb1 (s) + αb2 (s) ds s=0 ds s=0

= B(a, b1)w +T B (a, b2)w. Therefore the connection deﬁned by B is not only homogeneous but linear. As a consequence, the usual and most notable connections in Finsler geometry [2] (the Berwald, Cartan, Chern-Rund and Hashiguchi connections), which are homogeneous but nonlinear, can be properly and conveniently be deﬁned as linear ∗ connections on the pullback of the tangent bundle, with domain ˚τM (T M). ON THE LINEARIZATION OF A NON LINEAR CONNECTION 11

4. Natural prolongation The procedure described in this paper assigning a linear connection to a non linear one can be properly understood as a first order prolongation in the context of natural operations on fibered manifolds [12]. A systematic study of the functorial prolongation of a connection on a (non nec- essarily linear) fiber bundle B → M to a connection on the vertical bundle has been carried out in [8, 9, 10]. In [9] it is proved that a bundle functor G on the category of fibred manifolds admits a functorial operator lifting connections on B → M to connections on GB → B if and only if the functor G is a trivial bundle functor B 7→ GB = B × W , for some manifold W . As a consequence of this result, since the vertical functor is not a trivial functor, we deduce that there is no natural operator transforming connections on B → M to connections on Ver B → B. However, as it is already remarked in [9], the obstruction for the existence of such a natural operator may disappear if we consider some additional structure. In [12, §46.10], by restricting to the subcategory of affine bundles, it is shown that there exists a 1-parameter family of first order natural operators (natural on local isomorphisms of affine bundles) transforming connections on π : E → M into connections V on τE ≡ τE|Ver E : Ver E → E. ∗ ∗ V The pullback bundle π π : π E → E can be identified (via the vertical lifting ξπ) V with the vertical bundle τE : Ver E → E. Under this identification the linearized connection can be considered as a connection on the vertical bundle. Our linearization procedure transforms a connection on a vector bundle (a particular case of an affine bundle) into a linear connection on the pullback bundle, or equivalently to the vertical bundle. From the definition (Theorem 3.1) it is clear that our construction is a natural first order differential operator. Therefore it should coincide with one of the members of the above mentioned family. In [18] we provided an intrinsic expression for the members of that family. This expression was found by ‘brute force’ from the coordinate expression given in [12], but no geometrical understanding of the situation was provided. In this section we will find in more clear intrinsic terms the above family and we will prove that these are all natural first order prolongations of the given connection. We will try to keep it as elementary as possible, and we refer to the reader to [12] for further details of the theory of natural operators.

The functor that we are considering here is the pullback functor Pb from the category of vector bundles to the category of vector bundles. The functor Pb assigns the vector bundle Pb(E)= π∗E to a vector bundle π : E → M, and the morphism Pb(φ)=(φ,φ) to a vector bundle morphism φ: E → E′ between vector bundles. For every real number λ ∈ R we can deﬁne the family of maps Bλ by

V Bλ(a, b)w = B(a, b)w + λξπ(b, κ(w))

∗ for (a, b) ∈ π E and w ∈ TaE. It is easy to see that (a, b),w 7→ w, Bλ(a, b)w is an Ehresmann connection on π∗π, which will be denoted symbolically by Hor+ λκ V. Theorem 4.1: All ﬁrst order operators transforming a connection on a vector bundle into a connection on the pullback bundle which are natural on the local isomorphisms of vector bundles form the 1-parameter family Hor + λκV, for λ ∈ R. Proof. The diﬀerence between two connections on π∗E is of the form

H H ξ2 (a, b),w − ξ1 (a, b),w = w, B2(a, b)w − w, B1(a, b)w = 0, ∆(a, b)w 12 E. MART´INEZ for ∆(a, b) a linear map ∆(a, b): TaE → TbE with values in the vertical bundle. We V can write ∆ in the form ∆(a, b)w = ξπ(b, D(a, b)w) where D(a, b) is a linear map D(a, b): TaE → Eπ(a). H ∗ H Taking ξ2 an arbitrary connection on π E and ξ1 the linearized connection of Hor, ′ ′ V the associated maps B and B are related by B (a, b)w = B(a, b)w + ξπ b, D(a, b)w ∗ for linear maps D(a, b): TaE → Eπ(a) depending smoothly on the point (a, b) ∈ π E. Taking the horizontal and vertical components of w we can write w = ξH(a, v)+ V ξπ(a, c) with v = T π(w) and c = κ(w). Thus we have D(a, b)w = DH(a, b)v + H V DV(a, b)c, where DH(a, b)v = D(a, b)ξ (a, v) and DV(a, b)c = D(a, b)ξπ(a, c). ′ It follows that B is a natural operator if and only if both DH and DV are natural transformations, which means that for every local diﬀeomorphism (φ,ϕ) of E they satisfy

φ DH(a, b)v = DH φ(a),φ(b) Tϕ(v), φ DV(a, b)c = DV φ(a),φ(b) φ(c).

We now prove that these conditions imply that DH = 0 and that DV is a constant multiple of (a, b) 7→ idTπ(a)M . For that we just need to consider the trivial bundle pr1 : E = B×F → B, where B and F are Banach spaces. An element ((x, y), (x, z)) of π∗E will be written (x, y, z) ∈ B ×F ×F . The map DH will be of the form DH(x, y, z)(x, v)= x, D¯ H(x, y, z)v with

D¯ H(x, y, z) a linear map from B to F depending smoothly on (x, y, z). For every k ∈ R, the invariance under the morphism φ(x, y)=(kx,y) over the map ϕ(x)= kx reads D¯ H(kx, y, z)(kv) = D¯ H(x, y, z)v, and hence it implies D¯ H is homogeneous of degree −1 with respect to the variable x. Smoothness of D¯ H implies that it must vanish, hence DH = 0. The map DV will be of the form DV(x, y, z)(x, u)= x, D¯ V(x, y, z)u with D¯ V(x, y, z) a linear map from F to F depending smoothly on (x, y, z). For k ∈ R, we take the morphism φ(x, y)=(x,ky) over the identity φ(x) = x. The invariance condition for DV reads kD¯ V(x, y, z)u = D¯ V(x,ky,kz)ku, or in other words D¯ V(x, y, z)u = D¯ V(x,ky,kz)u. Thus D¯ V is homogeneous of degree 0 in the variables (y, z). Smooth- ness of D¯ V implies that it is independent on (y, z), and hence it depends only on the base point, i.e. DV(a, b) ≡ DV(m) with m = π(a)= π(b). For any linear automorphism A of F we consider the diﬀeomorphism φ(x, y) = (x, Ay). The invariance condition for DV reads AD¯ V(x)u = D¯ V(x)Au. Therefore the endomorphism D¯ V(x) commutes with any regular endomorphism, from where it follows that it is a multiple the identity. Thus DV(a, b) = α(m) idEm for some smooth function α ∈ C∞(M). Finally taking again the morphism φ(x, y)=(kx,y) we get α(kx)u = α(x)u, so that α is homogeneous of degree 0 and smooth, and hence constant, α(x)= λ. We conclude that D(a, b)w = λκ(w) for some constant λ ∈ R, and therefore ′ V ∗ B (a, b)w = B(a, b)w + ξπ(b, λκ(w)), for all (a, b) ∈ π E and all w ∈ TaE.

For λ = 0 we obtain the connection Hor, which is linear. For λ =6 0 the connection Hor + λκV is not a linear connection but an aﬃne connection on the bundle π∗E, with its canonical aﬃne structure, whose associated linear connection is precisely ON THE LINEARIZATION OF A NON LINEAR CONNECTION 13

Hor. Indeed,

′ ′ V ′ Bλ(a, b + b )w = B(a, b + b )w + ξπ(b + b , λκ(w))

′ d ′ = B(a, b)w +T B(a, b )w + (b + b + sλκ(w)) ds s=0 ′ V ′ = B(a, b)w +T B(a, b )w + ξπ(b, λκ(w)) +T 0b V ′ ′ = B(a, b)w + ξπ(b, λκ(w)) +T B(a, b )w +0b ′ = Bλ(a, b)w +T B(a, b )w, where in the ﬁrst step we have used linearity of B(a, b) in the argument b, and in the third one we have used the interchange law (see Section 2). Also, the linearized connection Hor is semibasic, while the other members in the family are not. Indeed, if w ∈ TaE is vertical then B(a, b)w = 0, and if we write V w = ξπ(a, c) for some c ∈ E, then

V V V Bλ(a, b)w = B(a, b)w + λξπ(b, κ(w))=0+ λξπ(b, c)= λξπ(b, c), which does not identically vanishes, except for λ = 0. We have proved the following statement. Proposition 4.2: The linearization Hor of a connection Hor can be characterized by any of the following properties: • Hor is the only natural ﬁrst order prolongation of Hor which is linear. • Hor is the only natural ﬁrst order prolongation of Hor which is semibasic.

In the ﬁnite dimensional case, in a local coordinate system the expression of the maps B for the connection Hor + λκV is

i ∂ A ∂ Bλ(a, b) w i + w A = ∂x ∂y (xi,yA)

i ∂ A A i A B i ∂ = w i + λ w − Γi (x, y)w − ΓiB(x, y)z w A . ∂x ∂y (xi,zA) It is clear that this connection is affine (the connection coefficients are affine functions of zA) and is linear only in the case λ = 0, which is the case of the linearization. The annihilator of the horizontal distribution is spanned by the 1-forms

A A B i A A i dz +ΓiB(x, y)z dx + λ dy − Γi (x, y)dx . 5. The covariant derivative In this section we will find an explicit expression of the covariant derivative associated to the linearization of an Ehresmann connection in terms of brackets of vector fields. As a preliminary result we have the following expression for the associated Dom- browski connection map. Proposition 5.1: The connection map κ¯ : T (π∗E) → π∗E for the linearized connection Hor is given by ′ ′ κ¯(w,w )= a, νπ w − B(a, b)w , ′ ∗ for (w,w ) ∈ T(a,b)(π E). 14 E. MARTÍNEZ

Proof. The vertical lift on the bundle π∗π : π∗E → E is given by

V V ξπ∗π (a, b), (a, c) = 0a,ξπ(b, c) . Indeed,

V d d V ξπ∗π (a, b), (a, c) = (a, b)+ s(a, c) = (a, b + sc) = 0a,ξ (b, c) . ds s=0 ds s=0

Therefore the vertical projection is νπ∗π(0a,w)=(a, νπ(w)) for w ∈ TbE vertical. The vertical projector P¯V for the linearized connection Hor is given by

′ ′ H¯ ′ ′ P¯V(w,w )=(w,w ) − ξ (a, b),w =(w,w ) − (w, B(a, b)w)=(0a,w − B(a, b)w). Hence

′ ′ ′ ′ κ¯(w,w )= νπ∗π(PV(w,w )) = νπ∗π(0a,w − B(a, b)w)= a, νπ(w − B(a, b)w) , which proofs the statement.

A section ζ ∈ Sec(π∗π) is of the form ζ(a)=(a, ζ(a)) where ζ : E → E is the corresponding section along π. From the result above we have that the covariant derivative of the section ζ in the direction of a vector w ∈ TaE is

Dwζ =κ ¯(Taζ(w))=κ ¯ w, Taζ(w) = a, νπ Taζ(w) − B(a, ζ(a))w , where m = π(a). From this expression it is clear that the formula for the covariant derivative is simpler when expressed in terms of sections along π. Therefore we will consider the covariant derivative as a map D : X(E) × Secπ(π) → Secπ(π) which is given by

Dwσ = νπ Taσ(w) − B a, σ(a) w , σ ∈ Secπ(π), w ∈ TaE. As a consequence of the above formula we have the following result. Theorem 5.2: The covariant derivative associated to the linearized connection Hor is given by

V DW σ = κ [PH(W ), σ ]+ T σ(PV(W )) for every section σ of E along π and every vector ﬁeld W ∈ X(E). Proof. We recall that on any manifold N the bracket [X,Y ] of two vector ﬁelds X,Y ∈ X(N) can be written in the form (see [12, §6.14])

ντN ◦ (TY ◦ X − χTN ◦ TX ◦ Y ) = [X,Y ], or equivalently V χ ξτN (Y, [X,Y ]) = TY ◦ X − TN ◦ TX ◦ Y.

In our concrete case the manifold N is N = E. For a basic vector ﬁeld X ∈ X(M) and a section σ along π we have

V V H V V H χ H V ξτE (σ , [X , σ ]) = T σ ◦ X − TE ◦ TX ◦ σ . (∗) ON THE LINEARIZATION OF A NON LINEAR CONNECTION 15

V H H Taking into account that νπ ◦ σ = σ and ξ (a + sb,X(m)) = X (a + sb), and using νTπ = T νπ ◦ χTE (Proposition 2.2 (2)), we have T σ(X H(a)) − B(a, σ(a))X H(a)=

H d H = T σ(X (a)) − T νπ ◦ χTE X (a + sσ(a)) ds s=0 V H H V = T (νπ ◦ σ )(X (a)) − T νπ ◦ χTE ◦ TX (σ (a )) V H H V = T νπ T σ (X (a)) − χTE ◦ TX (σ (a))

V V H V = T νπ ◦ ξτE σ (a), [X , σ ](a) V H V = ξπ σ(a), νπ([X , σ ](a)) , where in the last step we have used Proposition 2.1 (3), and we have taken into account that [X H, σV] is vertical because X is basic. Therefore H V (DXH σ)(a)= νπ [X , σ ](a) . For a vertical vector w ∈ Vera(π) we have B(a, b)w = 0, because Taπ(w) = 0. Hence Dvσ = νπ T σ(v) − B(a, σ(a))v = νπ T σ(v) . Up to now we have proved that H V V DXH+ηV σ = νπ [X , σ ]+ T σ(η ) , for η ∈ Sec(π∗E) and X ∈ X(M). In other words V DY σ = νπ [PH(Y ), σ ]+ T σ(PV(Y )) , for any projectable vector field Y∈ X(E). V For a general vector field W ∈ X(E) the bracket [PH(W ), σ ] is no longer vertical, and we cannot apply νπ directly. Projecting first to the vertical bundle and then applying νπ we have that the expression V V νπ PV([PH(W ), σ ]) = κ([PH(W ), σ ]) is C∞(T E)-linear in W , from where the result follows. Alternatively, one can check explicitly (as we did in [18]) that the expression on the right hand side of the formula in the statement satisfies the properties of a linear connection on π∗E. As every vector in T E can be obtained as the value of a projectable vector field the result follows.

Notice that a section σ along π is basic if and only if Dwσ = 0 for all vertical vector w. The expression for the covariant derivative in Theorem 5.2 was the starting point in [18]. In such paper the reader can ﬁnd further information about several applications of the theory.

6. Fibre derivative, curvature, and parallel transport It is clear from the formulas above that procedure of linearization of a connection has to do with the fiber derivative. In this section we will make precise this relationship. Given a (generally non linear) bundle map φ: E → F between two vector bundles π : E → M and τ : F → N fibered over a map ϕ: M → N the fibre derivative of φ is the vector bundle map Fφ: π∗E → τ ∗F fibered over φ determined by V F V ξτ ◦ φ = Tφ ◦ ξπ. 16 E. MARTÍNEZ

In more explicit terms, for (a, b) ∈ π∗E the fibre derivative of φ is of the form Fφ(a, b)=(φ(a), Faφ(b)) where Faφ is the restriction of Fφ to the fibre over a and F V is given by aφ(b)= ντ ◦ Tφ ξπ(a, b) . For the identity map in Ewe have F idE = idπ∗E. If ρ: G → P is a third vector bundle and ψ : F → G is another fibered map then F(ψ ◦ φ) = Fψ ◦ Fφ. These properties follow easily from the definition. We next show that the fibre derivative of a projectable vector field on a vector bundle defines a linear vector field on the pullback bundle. We recall [14] that a vector field Z ∈ X(F ) on a vector bundle τ : F → N is said to be linear if it satisfies one of the following two equivalent properties (1) The flow of Z is by vector bundle automorphisms. (2) Z is a vector bundle morphism from τ : F → N to T τ : T F → T N. There is a one to one correspondence between linear vector fields on τ : F → N and ∞ derivations of the C (M)-module Sec(F ). If {φs} is the local flow of a linear vector field Z then the derivation D associated to Z satisfies φ− η(ϕ (p)) − η(p) D d ∗ s s η(p)= φsη (p) = lim ds s=0 s→0 s where {ϕs} is the flow on the base (the local flow of the vector field in N to which Z projects). See [14] for the details.

We recall that if U ∈ X(P ) is a vector field on a manifold P with local flow {ϕs} C then the infinitesimal generator of the flow {Tϕs} is the complete lift U ∈ X(TP ) C of Z. It is related to the tangent of U by the canonical involution χTP ◦ T U = U . Proposition 6.1: Let Y ∈ X(E) be a projectable vector field. (1) The fibre derivative FY is a linear vector field on π∗E. (2) If {φs} is the local flow of Y then {Fφs} is the local flow of FY . Y (3) The derivation D : Secπ(E) → Secπ(E) associated to the linear vector field Y V FY is D σ = νπ [Y, σ ] . If Y ′ ∈ X(E) is another projectable vector field then (4) [FY, FY ′]= F[Y,Y ′], and ′ ′ (5) [DY , DY ]= D[Y,Y ]. Proof. From the definition of the fibre derivative we have that FY is a map F ∗ ∗ F ∗ Y : π E → (T π) (T E) projecting onto Y , i.e. pr1 ◦ Y = Y ◦ (π π). Moreover,

(τE, τE)(FY (a, b)) = τE(Y (a)), τE FaY (b) V = a, τE ◦ νTπ ◦ TY (ξπ(a, b)) V = a, νπ ◦ T τE ◦ TY (ξπ(a, b)) V = a, νπ(ξπ(a, b)) =(a, b), where we have used Proposition 2.1 (1). Under the canonical isomorphism T (π∗E) ≃ ∗ ∗ FY ∗ (T π) (T E) we have that (τE, τE) corresponds to π E / T (π E) ∗ τπ∗E, so that we have got a map FY : π E → ∗ ∗ ∗ π π T (π π) T (π E) such that τπ∗E ◦ FY = idπ∗E. Moreover, the Y ﬁbre derivative of a map is a vector bundle map, and E / T E thus we conclude that FY is a linear vector ﬁeld on π Tπ π∗E. This proves (1). M / T M X ON THE LINEARIZATION OF A NON LINEAR CONNECTION 17

We now prove (2). From the definition of the fibre V F V derivative we have ξTπ ◦ Y = TY ◦ ξπ. Applying the canonical involution to both sides we get V F C V Tξπ ◦ Y = Y ◦ ξπ, V V C where we have used χTE ◦ ξTπ = Tξπ (Proposition 2.2 (1)) and χTE ◦ TY = Y . F V C F Thus Y is ξπ-related to Y from where it follows that the local flow {Φs} of Y V V satisfies ξπ ◦ Φs = Tφs ◦ ξπ. This is the defining equation for the fibre derivative of φs, so that we conclude Φs = Fφs. For identifying the derivation associated to FY , for a ∈ E we have V F ∗ V F ξπ ( φs σ)(a) = ξπ φ−s(σ(φs(a))) V = Tφ−s σ (φs(a)) . V DY Computing ξπ( σ(a)) by means of the limit

V DY V 1 F ∗ ξπ( σ(a)) = ξπ lim [( φs σ)(a) − σ(a)] s→0 s

1 V V = lim [Tφ−s σ (φs(a)) − σ (a)] s→0 s V = LY σ (a),

Y V from where D σ(a)= νπ LY σ (a) . This proves (3). (4) and (5) are clearly equivalent, so that we will just prove (4). If {φs} is the flow ′ ′ ′ ′ of Y and {φs} is the local flow of Y then the commutator c(s)= φs ◦ φs ◦ φ−s ◦ φ−s ′ ′′ ′ satisfies c(0) = idE, c (0) = 0 and 2 · c (0) = [Y,Y ]. Applying the fibre derivative F ′ ′ F F ′ F F ′ we get (φs ◦ φs ◦ φ−s ◦ φ−s)= φs ◦ φs ◦ φ−s ◦ φ−s. Taking the second derivative with respect to s at s = 0 on the left hand side we get F[Y,Y ′] and on the right hand side we get [FY, FY ′]. Y For a projectable vector field Y the derivations D and DY are closely related but they are not equal. Proposition 6.2: If Y ∈ X(E) is a projectable vector field then Y D σ = DY σ − DσV (κ(Y )), for every section σ along π. Proof. If Y ∈ X(E) is projectable over X ∈ X(M) we can write Y = X H + ηV where η = κ(Y ) ∈ Secπ(E). Then Y H V V V D σ = νπ [X , σ ] + [η , σ ] , while H V V DY σ = νπ [X , σ ]+ T σ(η ) . V V V Notice that DηV σ = νπ(T σ(η )), from where it is easy to see that νπ([η , σ ]) = DηV σ − DσV η. Therefore the difference between both derivations is Y V V V V DY σ − D σ = νπ(T σ(η )) − νπ [η , σ ] = νπ T η(σ ) = DσV η, which proves the statement. It is apparent that the following class of vector fields will be very important in the description and analysis of the linearized connection. Definition 6.3: A vector field Y ∈ X(E) is said to be Hor-basic if it is projectable and κ(Y ) is a basic section. 18 E. MARTÍNEZ

In other words, a vector field Y ∈ X(E) is Hor-basic if it can be written as Y = X H +ηV with X ∈ X(M) (a basic vector field) and η ∈ Sec(E) (a basic section). The set of Hor-basic vector fields is a C∞(M)-module. From Proposition 6.2 we have Y that a vector field Y is Hor-basic if and only if D = DY . For a Hor-basic vector field Y on E the fibre derivative FY of Y equals the horizontal lift of Y with respect to the linearized connection. Theorem 6.4: The horizontal lift of a Hor-basic vector field Y with respect to the linearized connection Hor is equal to the fibre derivative of Y , that is Y H¯ = FY . Proof. For a horizontal projectable vector field Y = X H, with X ∈ X(M), we have H H H (FX )(a, b)= X (a), FaX (b) H H V = X (a), νTπ(TX (ξπ(a, b))) H d H = X (a), νTπ X (a + sb) ds s=0

H d H = X (a), νTπ ξ (a + sb,X(m)) ds s=0 H H = X (a), B(a, b)X (a) = ξH¯ (a, b),X H(a) . For a vertical vector ﬁeld Y = ηV, with η ∈ Sec(E) a basic section, using that νTπ = T νπ ◦ χTE on T π-verticals (Proposition 2.2 (2)), we have F V V V aη (b)= νTπ(T η (ξπ(a, b)))

d V = νTπ η (a + sb) ds s=0

d V = νTπ ξπ(a + sb, η(m)) ds s=0 d d = T νπ ◦ χTE a + sb + tη(m) ds dt t=0 s=0 d d = T νπ a + sb + tη(m) dt ds s=0 t=0 d V = T νπ ξπ a + tη(m), b dt t=0 d V = νπ(ξπ(a + tη(m), b)) dt t=0 d = b dt t=0

=0b, so that V V V V H¯ V (Fη )(a, b)= η (a), Faη (b) =(η (a), 0b)= ξ (a, b), η (a) . For a general Hor-basic vector ﬁeld Y we write Y = X H + ηV with both X and η basic, and hence by linearity of the horizontal lift and the ﬁbre derivative we have Y H¯ =(X H)H¯ +(ηV)H¯ = F(X H)+ F(ηV)= F(X H + ηV)= FY .

Parallel transport. As an immediate consequence of Proposition 6.1, if the local flow H¯ of a Hor-basic vector field Y is φs then the local flow of Y is Fφs. Thus for Hor-basic vector fields we have an explicit expression of the parallel transport system defined by the linear connection Hor, which complements the description given in [18]. ON THE LINEARIZATION OF A NON LINEAR CONNECTION 19

Theorem 6.5: Let Y ∈ X(E) be a Hor-basic vector field with local flow {φs}. For the linearized connection Hor the parallel transport map along a flow line s 7→ φs(a) of Y is Fφs. Conversely, if Hor is a connection on E and Hor′ is a connection on π∗E satisfying the above property then Hor′ is the linearization of Hor. ∗ Proof. The parallel transport of a vector (a, b) ∈ π E along γ(s)= φs(a) is given by the horizontal curve (with respect to Hor) projecting onto the curve γ with initial value (a, b), which is but the integral curve of Y H¯ starting at (a, b). Since Y H¯ = FY , such horizontal curve is but the integral curve of FY starting at (a, b), which is Fφs(a, b). Conversely, assume Hor′ is a connection on π∗E satisfying that the horizontal lifting of the curve s 7→ φs(a) starting at (a, b) is s 7→ Fφs(a, b) for every flow line ∗ φs of every Hor-basic vector field Y and every (a, b) ∈ π E. The tangent vector to the curve Fφs(a, b) at s = 0 is hence a horizontal vector d F F ′ φs(a, b) = Y (a, b) ∈ Hor(a,b) . ds s=0

On the other hand, from Theorem 6.4 we know that FY (a, b) ∈ Hor(a,b). As every tangent vector to E at the point a can be obtained by evaluating a Hor-basic vector ′ ∗ field at a it follows that Hor(a,b) = Hor(a,b). This being true for every (a, b) ∈ π E implies that the connection Hor′ and Hor are equal. For the vertical lifting Y = ηV of a basic section η ∈ Sec(E) we can provide an V explicit formula for the flow of FY . The flow of η is φs(a) = a + sη(m), with m = π(a). Thus

V d d V Tφs ξπ(a, b) = φs(a + tb) = (a + sη(m)+ tb) = ξπ a + sη(m), b , dt t=0 dt t=0 from where we get Fφs(a, b)=( a + sη(m), b). In other words, along any straight line contained in a fibre Em connecting two points a and a′ the parallel transport map is (a, b) 7→ (a′, b). This result holds true for any vertical curve (i.e. a curve contained in a fibre). Proposition 6.6: For the linearized connection, parallel transport along a vertical curve is the standard parallelism on the fibre (as a vector space). Proof. In fact we will prove that the result holds true for any semibasic connection ′ ∗ Hor in π E. Let γ :[t0, t1] → E be a vertical curve connecting the points a = γ(t0) ′ and a = γ(t1). A vertical curve γ(t) is entirely contained in a fibre, γ(t) ∈ Eπ(a). ′ ′ The horizontal lift γH of the curve γ with initial value (a, b) is γH (t)=(γ(t), b). ′ Indeed, the curve γH is horizontal

H′ dγ ′ (t) = (γ ˙ (t), 0 )= ξH (γ(t), b), γ˙ (t) ∈ Hor′ , dt b γ(t) because the connection is semibasic andγ ˙ (t) is a vertical vector, and obviously H′ H′ γ (t0)=(a, b). Therefore, the parallel translation of the vector (a, b) = γ (t0) is H′ ′ the vector γ (t1)=(a , b).

Curvature. We will find the curvature of the connection Hor. We denote by R the curvature 2-form of the nonlinear connection Hor. It is defined by H H H H H R(X,Y )= κ [X ,Y ] = νπ [X ,Y ] − [X,Y ] for X,Y ∈ X(M). 20 E. MARTÍNEZ

Theorem 6.7: The curvature of the linear connection Hor can be written as

Curv(Y1,Y2)σ = −DσV R(X1,X2)+ DXH η2 − DXH η1 1 2 H V for Hor-basic vector fields Yi = Xi + ηi , i =1, 2, and any σ ∈ Secπ(E). H V H V Proof. For Y1 = X1 + η1 , Y2 = X2 + η2 two Hor-basic vector fields we have H V H V κ([Y1,Y2]) = κ([X1 + η1 ,X2 + η2 ]) H V H V H V V V = κ [X1,X2] + R(X1,X2) + [X1 , η2] − [X2 , η1 ] + [η1 , η2 ] = R(X ,X )+ D H η − D H η , 1 2 X1 2 X2 1 V V V where we have taken into account that [η1 , η2 ] = 0 and DσV ηi = κ(T ηi(σ )) = 0, i =1, 2, because η1, η2 are basic, and Theorem 5.2. From Proposition 6.2 we have [Y ,Y ] D 1 2 V D[Y1,Y2]σ = σ + Dσ (κ([Y1,Y2])). From Proposition 6.1 (5) we have [DY1 , DY2 ] = D[Y1,Y2]. Thus for Hor-basic vector fields Y1 and Y2 and any section σ ∈ Secπ(E) Y Y [Y ,Y ] D 1 D 2 D 1 2 V [DY1 ,DY2 ]σ = [ , ]σ = σ = D[Y1,Y2]σ − Dσ (κ([Y1,Y2])). It follows that

Curv(Y1,Y2)σ = −DσV (κ([Y1,Y2])), H V and taking into account the expression of κ([Y1,Y2]) above for Yi = Xi +ηi , i =1, 2, we get

Curv(Y1,Y2)σ = −DσV R(X1,X2)+ DXH η2 − DXH η1 , 1 2 which proves the statement. As particular cases of the expression of the curvature we have

H H Curv(X ,Y )σ = −DσV R(X,Y ) V H Curv( η ,Y )σ = DσV DY H η Curv( ηV , ζ V )σ =0, for X,Y ∈ X(M), ζ, η ∈ Sec(E) and σ ∈ Sec(π∗E). In Finsler geometry, where E = T˚ M, the component θ(σ, η)Y = Curv( ηV ,Y H)σ is known as the Berwald curvature tensor, and the component Rie(X,Y )σ = Curv(X H,Y H)σ is usually known as the Finsler Riemannian curvature tensor. A detailed study of the curvature tensor of Hor and its implications for the original connection can be found in [18]. Here we just mention that the vanishing of the VH-component of the curvature tensor characterizes basic connections, that is, the linearization of a non linear connection is basic if and only if θ = 0. In what respect to the flatness of the linearized connection we have the following result. Theorem 6.8: The following statements are equivalent: (1) The linearized connection Hor is flat. (2) The set of Hor-basic vector fields is a Lie subalgebra of X(E). (3) If Y1,Y2 are Hor-basic vector fields then κ([Y1,Y2]) is a basic section. (4) If X1,X2 ∈ X(M) and η1, η2 ∈ Sec(E) then R(X1,X2)+ D H η2 − D H η1 is X1 X2 a basic section. ON THE LINEARIZATION OF A NON LINEAR CONNECTION 21

Proof. If Y1,Y2 are projectable then [Y1,Y2] is projectable. Therefore, for Y1,Y2 two Hor-basic vector ﬁelds the bracket [Y1,Y2] is Hor-basic if and only if κ[Y1,Y2] is basic. This proves the equivalence (2) ⇐⇒ (3). In the proof of Theorem 6.7 we found

κ([Y1,Y2]) = R(X1,X2)+ D H η2 − D H η1, X1 X2 from where it is clear the equivalence (3) ⇐⇒ (4). (2) =⇒ (1). For two projectable vector fields Y1,Y2 we know that [FY1, FY2] = F[Y1,Y2]. From Theorem 6.4, if Y1 Y2 and [Y1,Y2] are Hor-basic then this equation H¯ H¯ H¯ reads [Y1 ,Y2 ] = [Y1,Y2] . Since the horizontal distribution is spanned by Hor-basic vector fields it follows that Hor is an involutive distribution, and hence it is a flat connection. (1) =⇒ (3) follows from the expression of the curvature in Theorem 6.7. If Y1,Y2 are Hor-basic and the connection Hor is flat then Curv(Y1,Y2)σ = −DσV (κ([Y1,Y2])) = ∗ 0, for all σ ∈ Sec(π E). It follows that κ([Y1,Y2]) is a basic section, and hence [Y1,Y2] is Hor-basic.

7. The jet bundle approach An Ehresmann connection on a bundle τ : P → N can be equivalently seen as a 1 section of the first jet bundle τ10 : J τ → P . We refer to the reader to [27] for the notation and other details in jet bundle theory. 1 The 1-jet jnσ of a local section σ of τ at a point n ∈ N can be identified with the tangent map Tnσ : TnN → Tσ(n)P . Conversely, given a linear map φ: TnN → TpP such that Tpτ ◦ φ = idTnN there exists a local section σ of τ such that σ(n)= p and 1 Tnσ = φ. Therefore it make sense to write jmσ(v) for v ∈ TmM with the meaning 1 1 jmσ(v) ≡ Tmσ(v). The projection τ10 is defined by τ10(jnσ) = σ(n), and is said to 1 be the target projection. The composition τ1 = τ ◦ τ10 : J τ → N is said to be the source projection. If τ ′ : P ′ → N is another fiber bundle, a fibered map ψ : P → P ′ over the identity 1 1 1 ′ 1 1 1 in N induces a map j ψ : J τ → J τ fibered over ψ given by j ψ(jnσ)= jn(ψ ◦ σ). Given a connection on τ : P → N the corresponding horizontal lift ξH determines H a section h of τ10. Indeed, h(p) = ξ (p, · ): Tτ(p)N → TpP satisfies T τ ◦ h(p) = idTτ(p)N and hence it is a 1-jet with target p. Conversely, any section h of τ10 defines a connection on τ : P → N by means of the horizontal lift ξH(p, v) = h(p)(v). 1 Therefore a connection on τ : P → N is but a section of τ10 : J τ → P . 1 When τ : P → N is a vector bundle then the fiber bundle τ1 : J τ → N inherits a vector bundle structure with the operations 1 1 1 1 1 jnσ + jnη = jn(σ + η) and λ · jnσ = jn(λσ), 1 for σ, η ∈ Sec(τ) and λ ∈ R. The target map τ10 : J τ → P is a vector bundle morphism over the identity in M. A connection defined by a section h of τ10 is a linear connection if h a linear section of τ10, or in other words, if h is a vector bundle 1 morphism from τ : P → N to τ1 : J τ → N over the identity in N.

In our concrete case, given a non linear connection h ∈ Sec(π10) on the vector bundle π : E → M we know that the linearization of h is a linear connection on the ∗ ∗ ∗ vector bundle π π : π E → E, and hence it is given by a section h¯ ∈ Sec((π π)10) of the ﬁrst jet bundle J 1(π∗π) of sections of π∗π. In this section we provide a direct construction of h¯ by using the geometry of jet bundles. 22 E. MART´INEZ

Fh We can consider h as a (generally non linear) 1 E ×M E / J p 1 O O bundle map h: E → J π from the vector bun- 1 ≃ ≃ dle π : E → M to the vector bundle π1 : J π → M fibered over the identity in M. Thus we ∗ Fh ∗ 1 π E / π1(J π) can construct the fibre derivative obtaining a map ∗ ∗ 1 ∗ F π π pr1 h: π E → π1(J π). p h If we consider the bundle : E ×M E → M, the E / J 1π ∗ 1 1 manifold π1 (J π) can be identified with J p by iden- 1 1 1 π π1 tifying (jmσ, jmη) ≃ jm(σ, η). The projection onto 1 1 1 id the first factor (jmσ, jmη) 7→ jmσ is then identified M / M 1 with the first jet prolongation j pr1 of the projection pr1 : E ×M E → E onto the first factor. Thus 1 we obtain a map Fh: E ×M E → J p. Taking into account that π10 ◦ h = idE we have that Fπ10 ◦ Fh = F idE = idπ∗E. Moreover Fπ10 = p10, so that we have proved that Fh is a section of p10 and hence a connection on p: E ×M E → M. However this procedure does not produce a connection on π∗π : π∗E → E but a ∗ connection on the bundle p: E ×M E → M. In order to obtain a connection on π E we need a further step. The canonical immersion. As we know, a section of E determines a section of π∗E, which up to this point has been denoted by the same symbol. It is now convenient to distinguish between them. Consider the map ı: Sec(π) → Sec(π∗π) given by ı(σ)(a)= a, σ(π(a) , for all a ∈ E. The map ı induces a canonical immersion Υ: π∗(J 1π) → J 1(π∗π) by means of 1 1 Υ(a, jmσ)= ja ı(σ) . Interpreting a 1-jet with source m and target b as a linear map φ: TmM → TbE the expression of Υ is simply

Υ(a, φ)= idTaE,φ ◦ Taπ . The canonical immersion is a vector bundle map over the identity in E and, with respect to the target projection, it satisﬁes ∗ (π π)10 ◦ Υ = (pr1, π10 ◦ pr2), ∗ 1 or in more explicit terms (π π)10 Υ(a, jmσ) =(a, σ(m)), where m = π(a). Basic and semibasic connections can be characterized in terms of the canonical immersion Υ. ∗ ∗ Proposition 7.1: Let h ∈ Sec((π π)10) be a connection on π E.

• h is basic if and only if there exists a connection h ∈ Sec(π10) on E such that h(a, b)=Υ a, h(b) for all (a, b) ∈ π∗E. • h is semibasic if and only if Im(h) ⊂ Im(Υ). Proof. Let ξH the horizontal lifting associated to h, i.e. ξH((a, b),w)= h(a, b)(w). H H If ξ is basic then there exists a connection h ∈ Sec(π10) such that ξ ((a, b),w) = w,ξH(b, T π(w)) , where ξH is the horizontal lift associated to h. In terms of h and h this equation reads h(a, b)(w) = w, h(b)(T π(w)) , or in other words h(a, b) =

idTaE, h(b) ◦ Taπ = Υ(a, h(b)). Conversely, if hand h are related by h(a, b)=Υ a, h(b) , then for every w ∈ TaE we have ξH (a, b),w = h(a, b)(w)=Υ a, h(b) (w) = w, h(b)(w) = w,ξH(b, w) , so that the connection h is basic (and the pullback of the connection h). ON THE LINEARIZATION OF A NON LINEAR CONNECTION 23

∗ ∗ Let h ∈ Sec((π π)10) be a semibasic connection on π E, so that h(a, b)(w) = (w, 0b) whenever w ∈ Vera(π). It follows that h(a, b) is of the form h(a, b) =

(idTaE, Φ) for some linear map Φ: TaE → TbE such that Tbπ ◦ Φ = Taπ and Φ(w) = 0 for all w ∈ Vera(π). This implies that Vera(π) = Ker(Φ): obviously Vera(π) ⊂ Ker(Φ) and if w ∈ Ker(Φ) then Taπ(w) = Tbπ ◦ Φ(w) = 0, so that w ∈ Vera(π). Deﬁning the linear map φ: TmM → TbE by φ(v) = Φ(˜v) wherev ˜ is any vector in TaE such that Taπ(˜v) = v, we have that Tbπ ◦ φ = idTmM and Φ factorizes as Φ = φ ◦ Taπ. Thus φ is a 1-jet and h(a, b) = (idTaE,φ ◦ Taπ)=Υ(a, φ). Conversely, let h be a connection on π∗π such that Im(h) ⊂ Im(Υ). For every ∗ (a, b) ∈ π E there exists a map φb : TmM → TbE such that h(a, b) = Υ(a, φb) = H (idTaE,φb◦Taπ). If w ∈ TaE is vertical then ξ (a, b),w = h(a, b)(w)= w,φb(T π(w)) = (w, 0b). It follows that h is semibasic. ∗ 1 From the canonical immersion it is convenient to construct the map Ψ: π1(J π) → 1 ∗ 1 1 1 J (π π) given by Ψ(jmσ, jmη) = Υ(σ(m), jmη). The map Ψ is linear and satisﬁes ∗ (π π)10 ◦ Ψ=(π10, π10). Proposition 7.1 can be re-stated in terms of Ψ as follows: a connection h is basic if and only if there exists a connection h on E such that h(a, b)=Ψ(h(a), h(b)); a connection h is semibasic if and only if Im(h) ⊂ Im(Ψ). With the help of the map Ψ we can easily obtain the linearization of the connection h.

Theorem 7.2: Let h ∈ Sec(π10) be a non linear connection on E. The map h¯ =Ψ◦ ∗ Fh ∈ Sec((π π)10) is the section corresponding to the linearization of the connection h.

Proof. Using the linearity of π10 we have that 1 π10 Fah(b) = π10 lim [h(a + sb) − h(a)] s→0 s 1 = lim π10 h(a + sb) − π10 h(a) s→0 s 1 = lim [(a + sb) − a] s→0 s = b.

∗ It follows that the map h¯ is a section of (π π)10: ∗ ¯ ∗ (π π)10 h(a, b) =(π π)10 Ψ(Fh(a, b)) = π10(h(a)), π10(Fah(b)) =(a, b). Moreover, the connection h¯ is semibasic by construction (and Proposition 7.1) and a first order natural prolongation of h. It follows from Proposition 4.2 that h¯ is the linearization of h. Alternatively, one can easily check that h¯ is linear, because Ψ and Fah are both linear maps. Notice that the map B(a, b) associated to the linearization is given in this context by B(a, b)= Fah(b) ◦ Taπ. Remark 7.3: A construction for a kind of linearization of a non linear connection is given in [6, §2.7]. However, the result of the construction in [6] is not a connection ∗ on π E, but a connection on p: E ×M E → E (misleadingly identified there with the pullback bundle) which is identical to the connection Fh. The final step in [6], using 1 1 1 1 F the map ı0 : J π → J p, ı0(jmσ) = jm(σ, 0π), produces a map (a, b) 7→ ı0( ah(b)) which projects to (a, 0m), and hence it is not a connection on the pullback bundle π∗E. ⋄ 24 E. MARTÍNEZ

Construction on the vertical bundle. The pullback bundle π∗E is isomorphic to the vertical bundle Ver(π). Hence a construction of the linearized connection in the vertical bundle is possible. In this subsection we provide such construction, that is, given a non linear connection h: E → J 1π on π : E → M we will deﬁne a ¯ 1 V V linear connection hV : Ver(π) → J τE, on the vector bundle τE : Ver(π) → E which corresponds to h¯.

Consider the vector bundle pV : Ver(π) → M. We recall that there exists a 1 canonical isomorphism jvj: Ver(π1) → J pV deﬁned by

d 1 1 d jvj jmσs = jm σs , ds s=0 ds s=0 where σs is a curve in the set of sections of E. See [15, §5] for the details. 1 V Consider the connection h: E → J π. Given a vertical vector z = ξπ(a, b) ∈ 1 Vera π we consider the curve s 7→ h(a + sb) ∈ J π and we take the tangent vector 1 at s = 0. This is a vector on J π vertical over M (due to π1 ◦ h = π) to whom we 1 may apply the diﬀeomorphism jvj obtaining an element in J pV, V d 1 V h(z)= jvj h(a + sb) ∈ J pV, z = ξπ(a, b). ds s=0

It is easy to see that Vh is a linear connection on the bundle Ver(π) → M. Remark 7.4: Under the identification of E ×M E → M with Ver(π) → M given V F V by ξπ, the connection h corresponds to the unique natural prolongation h to a connection on Ver(π) → M defined in [12, §31.1]. Thus the connection Fh that we have obtained is the unique natural prolongation of h to a connection on p: E ×M E → M. ⋄ V A section V of pV : Ver(π) → M defines a section ıV(V ) of τE : Ver(π) → E by means of V ıV(V )(a)= ξπ a, νπ(V (π(a))) , for all a ∈ E. 1 1 V This relation induces a map ΨV: J pV → J τEgiven by 1 1 V V Ψ (jmV )= jτE (V (m)) ı (V ) . V V 1 The map ΨV satisfies (τE)10 ◦ΨV =(pV)10, or in other words (τE)10 ◦ΨV(jmV )= V (m) 1 1 for all jmV ∈ J pV. 1 V V In the following result j ξπ,E denotes the 1-jet prolongation of the map ξπ fibered 1 V V over the identity in E, and j ξπ,M denotes the 1-jet prolongation of the map ξπ fibered over the identity in M.

Theorem 7.5: Let h ∈ Sec(π10) be a non linear connection on E. The map ¯ 1 V ¯ V hV : Ver(π) → J τE deﬁned by hV = ΨV ◦ h is a linear connection on the ver- V ¯ 1 V ¯ ¯ V tical bundle τE : Ver(π) → E related with the connection h by j ξπ,E ◦ h = hV ◦ ξπ. Proof. The proof is based on the following facts whose proof is omitted. For the V 1 V jvj V canonical isomorphism we have that ◦ ξπ1 = j ξπ,M , and for the maps Ψ and Ψ 1 V 1 V ∗ we have the relation ΨV ◦ j ξπ,M = j ξπ,E ◦ Ψ. Then for all (a, b) ∈ π E,

¯ V d hV ξπ(a, b) =ΨV jvj h(a + sb) ds s=0 V V jvj F =Ψ ξπ1 ( h(a, b)) 1 V F =ΨV j ξπ,M h(a, b) 1 V F = j ξπ,E Ψ h(a, b) 1 V ¯ = j ξπ,E h(a, b) . ON THE LINEARIZATION OF A NON LINEAR CONNECTION 25

¯ ∗ V As h is a linear connection on π π and ξπ is an isomorphism of vector bundles it ¯ V follows that hV is a linear connection on τE.

Local coordinate expressions. In the ﬁnite dimensional case, consider a system i A i A A ∗ i A A 1 of local coordinates (x ,y ) on E, (x ,y , z ) on π E, (x ,y ,yi ) on J π, and i A A A A 1 ∗ (x ,y , z , zi , zB) in J (π π). The coordinate expression of Υ is i A i A A i A A A Υ (x ,y ), (x , z , zi ) =(x ,y , z , zi , 0), and hence the map Ψ is i A A i A A i A A A Ψ (x ,y ,yi ), (x , z , zi ) =(x ,y , z , zi , 0). i A i A A If h is given locally by h(x ,y )=(x ,y , −Γi (x, y)) then F i A i A i A A i A A B h (x ,y ), (x , z ) = x ,y , −Γi (x, y) , x , z , −ΓiB(x, y)z . Thus the composition of both maps gives ¯ i A A i A A A B h(x ,y , z )=(x ,y , z , −ΓiB(x, y)z , 0). In the case of the vertical bundle, we can take coordinates (xi,yA, zA) in Ver(π), i A A A A 1 i A A A A i A A A A (x ,y , z ,yi , zi ) in J pV, (x ,y ,yi , y˙ , y˙i ) in Ver(π1), and (x ,y , z , zi , zB ) in 1 V J τE, the local expression of jvj is i A A A A i A A A A jvj(x ,y ,yi , y˙ , y˙i )=(x ,y , y˙ ,yi , y˙i ). Due to the choice of the names of the coordinates, the local expression of Fh is exactly equal to the local expression of Vh, the local expression of ΨV is exactly equal to the local expression of Ψ, and the local expression of h¯V is exactly equal to the local expression of h¯.

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Eduardo Mart´ınez: IUMA and Departamento de Matematica´ Aplicada, Universi- dad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain

ORCID: https://orcid.org/0000-0003-3270-5681 E-mail address: [email protected]