Warps, grids and curvature in triple vector bundles

A triple vector bundle is a cube of vector bundle structures which commute in the (strict) categorical sense. A grid in a triple vector bundle is a collection of sections of each bundle structure with certain linearity properties. A grid provides two routes around each face of the triple vector bundle, and six routes from the base manifold to the total manifold; the warps measure the lack of commutativity of these routes. In this paper we first prove that the sum of the warps in a triple vector bundle is zero. The proof we give is intrinsic and, we believe, clearer than the proof using decompositions given earlier by one of us. We apply this result to the triple tangent bundle $$T^3M$$T3M of a manifold and deduce (as earlier) the Jacobi identity. We further apply the result to the triple vector bundle $$T^2A$$T2A for a vector bundle A using a connection in A to define a grid in $$T^2A$$T2A. In this case the curvature emerges from the warp theorem.


Double vector bundles, grids and warps
Double vector bundles arise naturally in Poisson geometry, in the connection theory of vector bundles, and generally in the study of geometric objects with two compatible structures. Double vector bundles have been "floating around" since at least Dieudonné's [4] treatment of connection theory, but the first systematic and general treatment was provided by Pradines [21]. A recent account with references is [16,Chap. 9]. We briefly recall the necessary facts.
A double vector bundle consists first of all of a square of vector bundles as shown in the first figure of (1). There are two vector bundle structures on D, with bases A and B, each of which is itself a vector bundle on base M; the two structures on D commute in the categorical sense (see below), and the map D → A × M B formed by the two bundle projections is a surjective submersion. We call this the double source map and denote it by ♮; that ♮ is a surjective submersion was proved by Li-Bland and Ševera [13]. 1 We further assume that ♮ has a right-inverse Σ : A × M B → D that is linear in both A and B; we call this a sigma-map.
The second and third figures in (1) show two standard examples arising from an arbitrary vector bundle A.IfA has a Poisson structure, then it is linear if and only if the associated map T * A → TAis a morphism of double vector bundles (in an obvious sense) of the structures above. When this is so, A is the dual of a Lie algebroid [3]. The third structure was introduced in global form in [19]. We give more details on this double vector bundle in Example 2. Double vector bundles also arise in the Lie theory of double Lie groupoids [17]; we will not consider this theory here. Each element d of a double vector bundle D may be represented in outline by the diagram in (2) which shows the projections of d under the two bundle projections. Given another element d ′ as shown, the sum over A has the outline shown in the third figure. db am , am .
(2) 1 We are grateful to Rajan Mehta for pointing this reference out to us.
The statement that the two vector bundle structures on D 'commute in the categorical sense' implies for the additions that where (d i ; a i , b i ; m), i = 1,...,4, have a 1 = a 2 , a 3 = a 4 , b 1 = b 3 and b 2 = b 4 . Equation (3)isthekeyinterchange law for double vector bundles. There are similar conditions involving the scalar multiplications, see [16, §9.1]. It follows that for elements d which project to zeros under both bundle projections, the two additions, and the scalar multiplications, coincide. Under these operations the set of such elements forms a vector bundle over M, called the core of D [21], usually denoted by C.
The core C is a submanifold of D; every element of C is an element of D. When working with examples, the core can usually be identified with a familiar vector bundle and it can be important to distinguish between elements of this bundle and the corresponding element of the double vector bundle. For example, the core of the double vector bundle TA, the middle diagram in (1), can be identified with A itself, [16, 9.1.7]. Therefore, an element a of the core A, can be viewed either as an element of A, or as an element of TA. In the latter case, we denote it by a ∈ TA. For general double vector bundles and triple vector bundles, this distinction is usually not necessary, so in Sects. 2, 3, and 4, we will not write bars over core elements. This distinction will be made clearly in Sect. 5. Now suppose that (d; a, b; m) and (d ′ ; a ′ , b ′ ; m) have a = a ′ and b = b ′ . Then there is a unique c ∈ C such that In equations of this type, what is important is that d − d ′ , calculated in either vector bundle structure of the double vector bundle D, results in the same core element c plus an appropriate zero. In Sect. 2.2, and particularly in Sects. 5 and 6, we will write such equations succinctly as d − d ′ ⊲ c. Expressions of this form are useful in summarizing detailed calculations. We find, however, that the ⊲ notation is not practical for detailed work; we have not attempted to develop a calculus for working with ⊲.
We now describe the original motivating example for the concepts of grid and warp.
In the 1988 edition of their book [1, p. 297], Abraham, Marsden and Raţiu gave the following formula for the Lie bracket of vector fields X and Y on a manifold M, where X is the complete lift of X to a vector field on TMand the uparrow denotes the vertical lift to T Y (m) TM of the vector [X, Y ](m). The complete lift, or tangent lift, X is J • T (X ) where J : T 2 M → T 2 M is the canonical involution which interchanges the two bundle structures on T 2 M. The double vector bundle T 2 M is a special case of the middle diagram of (1), where A = TM, and its core vector bundle is yet a third copy of TM. The left-hand side of (5) is encapsulated in (6).
If we look at the elements T (Y )(X (m)) and X (Y (m)), we see that they have the same outlines The two elements determine a unique core element c ∈ TM. Taking d = T (Y )(X (m)) and d ′ = X (Y (m)) in (4), we have . We adopt this notation whenever necessary, especially in Sects. 5 and 6.
Note that (5) needs to be proved in local coordinates, or in terms of the action of vector fields on functions. The use of (6) expresses the result in a compact conceptual way.
We now express these results in the terms that will be used throughout the paper. Consider a double vector bundle D as in (1). Definition 1 A pair of sections X ∈ Γ A and ξ ∈ Γ B D form a linear section of D if ξ is a morphism of vector bundles over X .
A grid on D is a pair of linear sections (ξ, X ) and (η, Y ) as shown in (7).
That nontrivial grids exist on a double vector bundle is ensured by the assumption that D → A × M B has a right-inverse that is linear in both A and B; that is, that D admits sigma-maps.
For each m ∈ M, ξ(Y (m)) and η(X (m)) have the same outline. They therefore determine an element of the core C and, as m varies, a section of C which we denote w(ξ, η). Precisely,
Note that w(ξ, η) changes sign if ξ and η are interchanged. Our convention gives the positive sign to the counterclockwise composition ξ • Y .
The question of signs-or orientations -will haunt us throughout the paper. Later on, we will see that there are various rules that, in many cases, determine which difference to take as the positive warp. These rules generally follow from established conventions of differential geometry.
Equation (5) can now be expressed as saying that the warp of (6)is[X, Y ]. Before proceeding, we give two further examples of grids and warps in double vector bundles.
Example 1 Consider the double vector bundle TA, the middle diagram in (1), where (A, q, M) is a vector bundle, and let ∇ be a connection in A.
Recall the horizontal lifting of vector fields from M to A induced by ∇. To define a vector field on A it is sufficient to define its effect on pullback functions f • q for f ∈ C ∞ (M), and on linear functions ℓ ϕ for ϕ ∈ Γ A * , defined by ℓ ϕ (a) = ϕ(q(a)), a .
Given a vector field Z on M, denote by Z H the horizontal lift of Z to A defined by where ∇ * is the connection in A * dual to ∇. Since Z H maps linear functions to linear functions and pullbacks to pullbacks, it is a linear vector field; it clearly projects to Z .
The word 'horizontal' here has its standard meaning in connection theory, and does not refer to the structures in TA.
It is straightforward to check that Now take any μ ∈ Γ A and form the grid shown in (9).

TA TM
AM .
We claim that the warp of the grid is ∇ Z μ; that is, for m ∈ M, where the right-hand side is the vertical lift of ( We give a sketch proof of (10). Again, it is sufficient to verify equality on linear functions and pullbacks. Applying each side of (10) to a pullback f • q, f ∈ C ∞ (M), gives zero.
Suppose given a sigma-map Σ : ThisZ H is a linear vector field over Z . Thus a sigma-map defines a horizontal lifting process, and thus a connection in terms of covariant derivatives ∇ Z . These processes can be reversed to show that every connection in terms of covariant derivatives ∇ Z defines a sigma-map.
It is thus possible to work with connections in vector bundles by using covariant derivatives, by horizontal lifting processes, or by using sigma-maps for the double vector bundle TA.The∇ formulation is now almost universal. However it does not relate easily to the notion of connection in a principal bundle. Kobayashi and Nomizu [11] gave two global definitions of a connection in a principal bundle P(M, G):asa suitable g-valued 1-form on P and as an invariant horizontal distribution on P.The latter defines, and is equivalent to, a lifting of vector fields on M to invariant horizontal vector fields on P, and this formulation resembles the lifting of vector fields in a vector bundle.
In the 1970s several authors, notably Dieudonné [4] and Besse [2], used the lifting formulation for connections in vector bundles. Dieudonné [4,XVII.16] used what we have called a sigma-map TM × M A → TA, and Besse used the corresponding left-split map TA→ A into the core. Equation (10) may be discerned on page 38 of [2] and is a special case of (17.17.2.1) in [4].
This example is central to Sect. 5. We will consider curvature in Sect. 5.
Example 2 For the third diagram in (1), first consider the manifold T * A where (A, q, M) is a vector bundle. There is a canonical diffeomorphism, denoted R,from T * (A * ) to T * A, which reverses the standard symplectic structures; see [19] and references given there. We define r : Givenasectionϕ ∈ Γ A * , the 1-form dℓ ϕ : A → T * A is a linear section over ϕ. Likewise given μ ∈ Γ A, we obtain a 1-form dℓ μ on A * . Composing with the map R : T * (A * ) → T * A, we obtain a linear section of T * A → A * over μ.

Outline of the paper
In a previous paper [18] one of us used a grid in the triple vector bundle T 3 M to express the Jacobi identity as a statement about the warps of the grids in the constituent double vector bundles. The proof given in that paper relied on a decomposition of T 3 M into seven copies of TM, and it was not clear whether the apparatus of grids and warps had provided a proof of the Jacobi identity or merely a formulation of it. One purpose of the present paper is to give an intrinsic proof of a general result for triple vector bundles and to resolve this question. Section 2 introduces the basic setup and notation for triple vector bundles. In Sect. 2.2 we formulate the main theorem of the paper on the warps of a grid on a triple vector bundle.
In a double vector bundle two elements with the same outline determine a core element (up to sign). In a triple vector bundle there are intermediate levels at which two elements may have the same outline, and there is more than one notion of core. Section 3 is concerned with describing the core elements determined by pairs of elements for which some levels of the outlines are equal. A brief summary is given at the end of the section. Section 4 gives the proof of the warp theorem, Theorem 1. This states, roughly, that given a grid on a triple vector bundle E, the sum of the ultrawarps is zero. The ultrawarps are the warps of the grids induced on the core double vector bundles by the grid on E. The proof is intrinsic and does not rely on a decomposition of E.Abrief summary is given at the end of the section.
In Sect. 5 we consider a vector bundle A and the triple vector bundle T 2 A.A connection in A induces a grid in T 2 A, and we show that the concept of curvature arises from the warp theorem; see Theorem 2. Section 6 presents the example of T 3 M and the deduction of the Jacobi identity for the Lie bracket of vector fields on M from the warp theorem.
Some concluding remarks are given at the end of the paper.

Triple vector bundles and the warp theorem
In this section we do two things. First, we set up everything we need for triple vector bundles, in order to formulate the warp theorem (Theorem 1). Secondly, we describe the original formulation [18] of the theorem, and outline the steps which lead to an intrinsic proof.

Basics on triple vector bundles
The basic structure of triple vector bundles has been given in [9,15,18]. Consider the following diagram, where each edge is a vector bundle. By an upper face we mean a face which has E 1,2,3 as total space. The lower faces are the three faces which have M as base manifold. We refer to the faces by the names Back, Front, Left, Right, Up, Down.
The total space of (11) should be denoted, for consistency with the labeling scheme, by E 1,2,3 but we will usually denote it by E.

Definition 3
A cube of vector bundles is a system of vector bundle structures as in (11), such that each face is a double vector bundle, and such that the vector bundle operations in E → E 1,2 are morphisms of double vector bundles from the Up face of E to the Down face of E and similarly for the other vector bundle structures in E.
The terminology 'cube of vector bundles' is temporary; in [9,15,18] the definition of a triple vector bundle included the (sometimes tacit) requirement that what was there called a decomposition existed, and this can be seen to be equivalent to the existence of a sigma-map. In order to isolate the condition that ensures the existence of a sigmamap, we introduce the above terminology for structures without this condition. In the rest of this subsection we work with such a single cube of vector bundles E.
How do we add elements in E?I fe, f ∈ E lie over the same point of E 2,3 ,a s shown in (12), their sum has the outline shown in (13 The outlines for scalar multiplication are similar. Core double vector bundles and the ultracore. Since each face of E is a double vector bundle, each face has a core vector bundle. The cores of the lower faces E i, j are denoted E ij with the comma removed. The core of the upper face with base manifold E k is denoted E ij,k . (This convention comes from [8].) Focus on the core vector bundles of the Up and of the Down faces. The Up face projects to the Down face via the double vector bundle morphism which consists of the bundle projections E 1,2,3 → E 1,2 , E 2,3 → E 2 , E 1,3 → E 1 and E 3 → M.The restriction of E 1,2,3 → E 1,2 to E 12,3 goes into E 12 and inherits the vector bundle structure of E 1,2,3 → E 1,2 . Together with the vector bundle structures on the cores of the Up face and the Down face, this yields another double vector bundle, with total space E 12,3 , which we call the (U-D) core double vector bundle.
Of course this can also be done for the other two pairs of parallel faces. So there are three core double vector bundles, shown in (14).
Elements of the core of E 12,3 project to zeros in the Down face. In the Up face they project to zeros over the zero in E 3 . It follows that an element of the core of E 12,3 projects to zero in every bundle structure. Equally the cores of the (B-F) and (L-R) double vector bundles consist of the elements of E 1,2,3 which project to zeros in every bundle structure. Thus each double vector bundle in (14) has the same core. This is denoted E 123 (without commas) and called the ultracore of E.
From the interchange laws it follows that the three additions on E, namely + , coincide on the ultracore and give it the structure of a vector bundle over M.
Notation for zero sections. The zero section of E 1 is denoted by 0 The double zero of E 1,2 is denoted by ⊙ 1,2 m , with similar notations for the other vector bundle structures.
The zero section of E → E 1,2 is denoted by0 : E 1,2 → E, e 1,2 →0 e 1,2 . Note that the subscripts of the element e 1,2 are enough to indicate that this is the zero section of E over E 1,2 ; there is no need for superscripts on0.
Finally, the triple zero of E is denoted by ⊙ 3 m . This is the zero of the ultracore vector bundle.
Notation for projections. The superscripts of the projection maps of a cube of vector bundles E denote the domain, and the subscripts denote the target, for example, q 1,2 2 : E 1,2 → E 2 . We omit the corresponding script when the domain is E, for example q 1,2 : E → E 1,2 , or the target is M, for example q 2 : E 2 → M.
Instances of the interchange law. As mentioned in the Introduction Eq. (3) is one of the interchange laws for double vector bundles. In the triple vector bundle setting, since E has three different vector bundle structures, we have interchange laws relating either two, or all three vector bundle structures. In this paper we only need the interchange laws that relate two of the three vector bundle structures.
Without loss of generality, focus on the Left face. This is a double vector bundle. The interchange law in the Left face for four elements e, f, g, h ∈ E, over the same e 2 ∈ E 2 is the following, where the outlines of the four elements in question are: Recall that the projection map from the Left to the Right face is a double vector bundle morphism. Consequently, the interchange law in the Right face holds for the projections of the four elements e, f, g, h ∈ E in E 1,3 : namely e 1,3 , f 1,3 , g 1,3 , and h 1,3 : with outlines

Result of Li-Bland and Ševera for cubes of vector bundles
In [13] Li-Bland and Ševera proved that the condition in the definition of a double vector bundle D on page 2, that the double source map ♮ : D → A× M B is a surjective submersion, actually follows from the other conditions. We now prove the corresponding result for cubes of vector bundles. We first need to show that the set that corresponds to A × M B is a manifold.

Lemma 1 Given a cube of vector bundles E, write W for the set of all
Then W is a closed embedded submanifold of E 1,2 × E 2,3 × E 1,3 .
Note: This result is valid for three double vector bundles arranged as are the three lower faces of E. In the terminology of [15], the result holds for cornerings.
This is (a rearrangement of) the product of three surjective submersions. Define This Δ is a submanifold of the target of F and F is a surjective submersion, so then all three elements project to the same element of M.Given(e 1,2 , e 2,3 , e 1,3 ) ∈ F −1 (Δ), write e 1 , e 2 , e 3 as above and write m = q 1 (e 1 ). Then This completes the proof of the Lemma.
The triple source map of the cube of vector bundles E, denoted ♮,i st h em a p E → W formed from the bundle projections of E; that is, ♮(e) = (q 1,2 (e), q 2,3 (e), q 1,3 (e)).
Now let us denote by the double source maps for the Back face and for the Front face of E, respectively. (11), then the triple source map

Proposition 1 If E is a cube of vector bundles as in
The key to the proof is the following commutative diagram. Note that ♮ B is a morphism of vector bundles over ♮ F ;(17) is not a double structure.
Take any e 2,3 ∈ E 2,3 , with ♮ F (e 2,3 ) = (e 2 , e 3 ). Using the notation for the zero section of E → E 2,3 as introduced earlier, write0 e 2,3 ∈ E e 2,3 . The decompositions of the following tangent spaces are a standard result of vector bundle theory: Take the tangent of the vector bundle morphism (17) at point0 e 2,3 . This decomposes into .
By applying the tangent functor to (11) (we do not need to consider the hypercube structure), and applying the same argument to the resulting structure, we see that the triple source map is a submersion.
This completes the proof of Proposition 1.
The result of Li-Bland and Ševera on which Proposition 1 is based was established for VB-Lie groupoids rather than double vector bundles. By combining the techniques of their proof and that above, we could establish a similar result for VB-double Lie groupoids.
We can now state the definition of a triple vector bundle.

Definition 4 A triple vector bundle is a cube of vector bundles for which there exists a morphism of cubes of vector bundles
It seems likely that the existence of sigma-maps follows from the definition of a cube of vector bundles (compare [6]) or by using Proposition 1, but we prefer to state it as an explicit assumption.
Grids on triple vector bundles. A grid in a double vector bundle constitutes two linear sections. In a triple vector bundle the concept of grid requires what we call linear double sections.

Definition 5 A down-up linear double section of E is a collection of sections
which form a morphism of double vector bundles from the Down face to the Up face.
The core morphism of Z 1,2 defines a vector bundle morphism from the core of the Down face to the core of the Up face. We denote this by Z 12 : In a similar fashion we define right-left and front-back linear double sections of E.

Definition 6
A grid on E is a set of three linear double sections, one in each direction, as shown in (21).
That nontrivial grids exist is guaranteed by the existence of sigma-maps; more precisely, a grid exists for any given sections X , Y , Z .
We note the following equations for future reference. They follow from the fact that the double sections are morphisms of double vector bundles.
For e 1,2 , e ′ 1,2 over the same point of E 1 , For e 1,2 , e ′ 1,2 over the same point of E 2 ,

Formulation of the warp theorem
We now have everything we need in order to describe the original formulation of the theorem, as given in [18]. Start with a grid on E. Focus on the Up face of the triple vector bundle. We see that (Y 1,3 , Y 3 ) and (X 2,3 , X 3 ) define a grid on the Up face. Denote its warp by w up . This is a section of the core vector bundle of the Up face, that is, w up : Similarly for the Down face, its warp w down is a section of the core vector bundle of the Down face, so w down : M → E 12 .Itfollowsthat(w up , w down ) is a linear section of the (U-D) core double vector bundle. Recall that the core morphism Z 12 of the linear double section Z 1,2 defines another linear section of the (U-D) core double vector bundle. Therefore, we have the following grid on E 12,3 : We call the warp of this grid the Up-Down ultrawarp and denote it by u UD .I ti sa section of the ultracore E 123 . Of course we can also build corresponding grids on the other two core double vector bundles. We therefore have three ultrawarps, as shown in (24).
We take the ultrawarps with the orientations opposite to (24); that is, using the ⊲ notation, We can now state the main theorem about grids on triple vector bundles. (21),

Theorem 1 (Warp Theorem) Given a triple vector bundle E and a grid on E as in
To give an intrinsic proof, we need to describe the ultrawarps in an alternative way. So far, the only equation we have seen that describes the warp of a grid on a double vector bundle is (8). Focus on the ultrawarp u UD . From the grid on the (U-D) core double vector bundle, for m ∈ M,by (8) we have that How can we express (w up •Z )(m) and (Z 12 • w down )(m) i nam o r eu s e f u lw a y ? About w up , for any e 3 ∈ E 3 ,againfrom(8) we have that Putting We introduce a more succinct notation, for use in calculations.
where e ′ 1,3 = X 3 (Z (m)) and λ 3 = w up (Z (m)). Note the following. In the case of a double vector bundle, we can rewrite (8)as In the case of a triple vector bundle, this is also possible. If we tried a similar calculation on (30), since0 e ′ 1,3 - Apply Z 1,2 to this, and using (22) and (23) it follows that Again, for reasons of economy of space, rewrite this as where e 1,3 = Z 1 (X (m)) and k 3 = Z 12 (w down (m)). Alternatively, as in the case of λ 3 , Let us go back to (27). We can rewrite this as and using (31)  or, more elegantly, using interchange laws, In calculations it is generally preferable to use equations of the form (30), and to avoid equations of the form (31).
Therefore, in order to describe ultrawarps such as u UD (m), we will use equations of the form (33), and in abbreviated notation, as introduced after (4).
It is worth emphasizing that the above arguments rely on the fact that core and ultracore elements are uniquely determined by equations such as (8).
There are similar abbreviated equations for the other two ultrawarps. Altogether we have and from now on we will use a further shortening of the notation, The main difficulty in proving (26) is that we cannot simply add and subtract the expressions in (34), since the operations are in different vector bundle structures. The apparatus of the next section overcomes this difficulty.

Orientation.
A further problem arises from the fact that the warp of a grid on a double vector bundle is only defined up to sign. We now need to consider how to choose these signs consistently for a grid on a triple vector bundle. This is a question of fixing orientations.
We choose to orient each upper face so that the positive term in the formula for the warp defines the outward normal by the right-hand rule. We then take the positive and negative terms in the opposite lower face to match those in the upper face; that is, we orient the lower faces so that the positive term in the warp defines the inward normal.
Thus the orientation of the Up face determines the signs in the first subtraction in (34c) below and the orientation of the Down face determines the signs in the second subtraction.
The "middle subtractions" in (34), that is, the orientations of the core double vector bundles, is an independent choice, equivalent to the choice of signs in (25). What matters here is consistency: if we took all three ultrawarps with the opposite signs, that would also be fine.

Preliminaries for the proof of the warp theorem
This section contains the main technical work needed for the proof of the warp theorem. We first describe our approach.
We want to find how the three ultrawarps are related, more specifically, for m ∈ M, we want to find a relation between three ultracore elements, u 1 , u 2 , u 3 . The best way to do this, where slightly easier calculations are involved, is to manipulate the differences of the six elements, (34a), (34b), and (34c). That is exactly what we do in this and the following section. In this section we obtain formulas expressing the difference of two elements of a triple vector bundle in terms of core elements and zeros. If two elements of a triple vector bundle can be subtracted, then their outlines must have at least one face in common. Cases where the outlines have two or more faces in common arise repeatedly in the rest of the paper. Each of these cases needs individual treatment.
Looked at from another point of view, two elements of a triple vector bundle which can be subtracted may admit exactly one, or two, or all three, of the subtractions - Step 3. Applying double vector bundle theory again, we get where u 1 is an ultracore element. where k 2 ∈ E 13,2 , core of the Left face, with outline So, we can write where u 2 is an ultracore element. Similarly for the Up faces, we have e 3 with u 3 an ultracore element.
Step 5. Show that u 1 = u 2 = u 3 . We show that u 1 = u 3 . So far, we have two expressions for e -1,3 e ′ , namely: Expand the left-hand side of (35), mimicking the double vector bundle case: Therefore, we see that (35) can be rewritten as: from where it follows that u 1 = u 3 . Similarly, we can show that u 2 = u 3 . At this point write u 1 = u 2 = u 3 to be u.
Step 6. We obtain six formulas for the differences between e and e ′ .

u).
What is important here is that the subtraction with respect to each structure results in the same ultracore element u.
We will use the following special case later on.
and therefore0

Second case: two elements that have two lower faces in common
What happens if e and e ′ have only two of the lower faces in common? Then only two of the three subtractions are defined. There are three cases to consider, each of which arises later.
If e and e ′ have the same Front and Right face. Since e and e ′ have the same e 1,3 and e 2,3 , it follows that they have the same e 1 , e 2 and e 3 . However e and e ′ will differ at e 1,2 and e ′ 1,2 , and these will differ by a core element w 12 ∈ E 12 of the core of the Down face, that is where k ∈ E 23,1 , the core of the Up face. Also, using the morphism q 1,2 : E → E 1,2 , we show that q 1,2 (k) = w 12 . First, where k is an element of the core of the Back face with outline

Special case: differences between zero elements
Using the fact that0 is the zero section 0 e 2,3 + There are similar formulas in the other two cases.

Summary of Sect. 3
-Given two elements e, e ′ ∈ E with the same outline, then they differ by a unique ultracore element u ∈ E 123 : ee ′ ⊲ u.
-Given two elements e, e ′ ∈ E with two lower faces in common, then their difference ee ′ ⊲ k, defines a k in one of the core double vector bundles: -i fe, e ′ have the same Right and Down face, then k ∈ E 23,1 , -i fe, e ′ have the same Front and Down face, then k ∈ E 13,2 , -i fe, e ′ have the same Front and Right face, then k ∈ E 12,3 .

Notation
In this section we prove Theorem 1. We will use the notation of (29). We further simplify the notation for elements of the lower faces and edges, as follows  We will need the following relations for the core elements of the lower faces in detailed form.
where w 23 ∈ E 23 , w 13 ∈ E 13 and w 12 ∈ E 12 . For the zeros of these w elements, the diagrams arê

Core and ultracore elements arising from the grid
We collect here for reference the definitions and outlines of the core and ultracore elements arising from the grid. YZX =0 e 1,2 + 1,3 If we look at XZY and XYZ, we see that they also have two faces in common, and their differences define a k 1 ∈ E 23,1 , with outline The two differences are, again using (40), • λ 2 , k 2 and u 2 . The same procedure can be applied to XZY and ZXY; they have the same Front and Down faces, so their differences will define an element λ 2 ∈ E 13,2 The corresponding equations, using (39), are If we look at YXZ and YZX, their differences define a k 2 ∈ E 13,2 , with outline YZX =0 e 1,2 + 2,3 Since λ 2 and k 2 have the same outlines, they differ by an ultracore element u 2 ∈ E 123 , (0 e 1 + 2,3 • λ 3 , k 3 and u 3 . Likewise YXZ and XYZ define λ 3 ∈ E 12,3 with outline The corresponding relations are, due to (38), Likewise ZYX and ZXY define a k 3 ∈ E 12,3 with outline The differences defined are, due to (38), The four relations in this case are

Proof of the warp theorem
We will show that u 1 + u 2 + u 3 =⊙ 3 m by showing that u 1 =− u 2 − u 3 . There are four steps.
Step 1. Rewrite (49b), using the double vector bundle interchange law in the Left face. We know from (49b) that the ultracore element defined by the first expression is u 1 , therefore, the ultracore element of the latter expression will also be u 1 . We will show that the second expression has −u 2 − u 3 as its ultracore element, and this will show that u 1 =−u 2 − u 3 . Step Step 4. Since our convention is that λ 3 − k 3 defines u 3 ,i tf o l l o w st h a tk 3 − λ 3 defines −u 3 . It is essential to follow these conventions.
We are finally able to complete the proof of Theorem 1. First, using operations in E → E 2,3 ,wehave Now, using (54) and (55), this is equal to Applying the interchange law in the Left face to the outer operations, this becomes Applying the interchange law in the Back face, in each [],wehave (0 e 2 + 1,30 Rewrite this as : and this is equal to0 [0 e 2 -2,30 using the facts that the zeros0 w 23 in the first []are zeros over E 2,3 , and then the interchange law in the Up face. Likewise, using the fact that the zeros0 e 2 are zeros over E 2,3 , this is equal to0 Finally, using an equation of the form (37), this becomeŝ 0 w 23 + 1,30 from which we obtain −(u 3 + u 2 ) as the ultracore element.
Comparing this with (49b), This completes the proof of the warp theorem.

Summary of Sect. 4
AgridonE,form ∈ M, defines six elements, ZYX, YZX, XZY, ZXY, YXZ, XYZ. These elements define the following core and ultracore elements: In the final step of the proof we showed that and since the left-hand sides are equal, it follows that u 1 =−u 2 − u 3 .

Remark 1
The strategy of this proof deserves some commentary. What should the warp of a grid on a triple vector bundle be? Or, in other words, why are we interested in the ultrawarps of a grid of a triple vector bundle?
The warp of a grid in the double case is a section of the core vector bundle, and measures the non-commutativity of the two routes defined by the grid.
So far, we have seen that all operations on a triple vector bundle are iterations of operations defined in double vector bundles. The ultracore, for example, is the core of the core double vector bundles.
For these reasons, we would want the warp of a grid in the triple case to be a section of the ultracore vector bundle, and to measure the non-commutativity of routes defined by the grid.
Pick an upper face of E, for example the Up face. If we compare the two routes defined by the grid in this face, then we obtain an element of the (U-D) core double vector bundle, which we denoted by λ 3 . Similarly for the other upper faces, the non-commutativity of the corresponding routes defines λ 1 and λ 2 .Thethreeλ's are elements of different spaces; therefore, if we tried to compare them, or indeed perform any sort of operation with them (such as adding them or subtracting them), we would see that such an operation could be algebraically possible but would not be geometrically meaningful.
The same applies for the three k i defined by the comparison of the routes for the lower faces.
The λ i 's and the corresponding k i 's, however, are elements of the same spaces, and so comparing them is a possibility, and indeed the only sensible operation. And by comparing them, we measure the non-commutativity of four routes, instead of two.
This can be done for the three pairs of λ i and k i , and so we obtain the three ultrawarps.
So what does the warp theorem tell us? Each ultrawarp measures the non-commutativity of four routes. In total, a grid on a triple vector bundle provides six different routes from M to E. The sum of the three ultrawarps takes into account each route twice, once with a positive and once with a negative sign. (Compare the description of curvature identities in terms of paths around oriented faces in cubes and hypercubes given by [20,Chap. 15].) The warp theorem tells us that these add up to zero, a result that seems reasonable. The different vector bundle structures over which the operations take place, however, are the main obstacle here-as soon as one realizes that simple operations like addition and subtraction in the triple vector bundle setting are no longer simple. Further analysis of grids in triple vector bundles is given in [5].

The triple vector bundle T 2 A and connections in A
In Example 1 of the Introduction we started with a vector bundle A → M and a connection in A, and we saw that the warp of the grid (9)i s∇ Z μ,t h e covariant derivative of μ with respect to Z . In this section, starting with a vector bundle A → M and a connection in A, we will build a grid on the triple vector bundle T 2 A. We now apply the warp theorem to this grid, and we thereby obtain a new proof of the equivalence of the definition of curvature in terms of horizontal lifts and the definition in terms of covariant derivatives (see Theorem 2). (A, q, M) there is a triple vector bundle structure on T 2 A as shown in (56).

For a vector bundle
Here the Down face is the usual double vector bundle TA and the Up face is the tangent prolongation of this; that is, it is obtained by applying the tangent functor to each structure in TA. Each vertical vector bundle in (56) is a standard tangent bundle.
In the first subsection we describe in detail the core double vector bundles of T 2 A. In the second subsection we investigate J A : T 2 A → T 2 A, the canonical involution of the manifold A, as a morphism of triple vector bundles. In the third subsection we build the aforementioned grid on T 2 A, and in the two remaining subsections we calculate the three ultrawarps induced by this grid.

The core double vector bundles of T 2 A
The three core double vector bundles of (56) are shown in (57), in the usual order (B-F), (L-R), and (U-D), and arranged as in (14).
where t · TM ξ denotes scalar multiplication in T (q) : TA→ TM.
The Left face is the double tangent bundle of the manifold A.Gi v enξ ∈ TAthe core element is where the scalar multiplication is in the usual tangent bundle TA→ A.
For the core of the Up face, first write ξ = d dt a t t=0 where a t is a curve in A. Write a t ∈ TAfor the core element corresponding to a t . Then . (60)

The canonical involution on T 2 A
The canonical involution J A : T 2 A → T 2 A for the manifold A is an isomorphism from the double vector bundle T 2 A to its flip. In what follows we will need to use it as a map of triple vector bundles.
Consider now the maps which J A induces on the cores. Take an element ξ ∈ TAin the core of the Back face. Regarded as an element of T 2 A this is ξ B , with outline shown on the left of (62).
It follows from (58) and (60) that since J 2 A is the identity. Since the Left faces in (61) are the double tangent bundle T 2 A,t h em a po nt h e cores of the Left faces is the identity and so

Grids on T 2 A
Now consider a connection ∇ in A. Example 1 gave a construction of a grid in TA for which the warp is ∇ X μ. We now extend this idea to define a grid in T 2 A. Let X, Z ∈ X(M), and μ ∈ Γ A. Define the following three linear double sections: -From Front to Back face: (T (X H ); X H , T (X ); X ).
-From Down to Up face: ( Z H A ; Z , Z H ; Z ).
Here Z = J M • T (Z ) is the complete (or tangent) lift of Z to a vector field on TM.
Likewise Z H A is the complete lift of Z H ∈ X(A) to a vector field on TA. The grid is shown in (65).
The core morphisms of the linear double sections will be needed later: The first two cases are instances of the general fact that given a morphism (ϕ, f ) of vector bundles, the core morphism of the double vector bundle map (T (ϕ); ϕ, To calculate the core morphism of ( Z H A ; Z , Z H ; Z ), focus on (66). At this point we investigate this linear double section further; it is a double vector bundle morphism from the Down face to the Up face of T 2 A. Note that (66) is not a triple vector bundle.
Note the following. Initially, a ∈ TA is in the core of the Down face of T 2 A. The canonical involution J A maps the Down face to the Front face (see (61)). Therefore, in T (Z H )(a), a is now an element of the core of the Front face. The maps (T (Z H ); Z H , T (Z ); Z ) form a double vector bundle morphism from the Front to the Back face of (56), with core morphism (Z H , Z ) as usual. Therefore, is now in the core of the Back face. And by (63), it follows that This completes the proof that the core morphism of ( Z H A ; Z , Z H ; Z ) is (Z H , Z ).

The six warps of grid (65)
The first step in applying the warp theorem to the grid (65)onT 2 A was to calculate the core morphisms of the linear double sections of the grid; we have done this in the previous subsection. Now, for each face of T 2 A, we find the warp of the grid defined by (65). We present the six warps in Table 1,usingthe⊲ notation introduced directly after Eq. (4), on page 3. The three warps w front ,w right , and w down of the lower faces follow directly from Example 1 and Eq. (5). The warps of the upper faces are a bit trickier.
We begin with the warp of the Up face. For this, we need the following proposition which states that the warp of the tangent of a grid is the tangent of the warp of the grid.  T (ξ ) Proof We leave the verification that TD is a double vector bundle and that (T (ξ ), T (X )) and (T (η), T (Y )) are linear sections of TDto the reader. The warp w(ξ, η) ∈ Γ C of (ξ, X ), (η, Y ) is given as usual by (8). We calculate the warp of the tangent grid. From the definition of a warp, for x ∈ T m M, ,form t a curve in M with tangent vector x at t = 0. Then, for Here we used the formula for the addition in TD→ TA, and the corresponding formula for scalar multiplication.
In (69), where F ∈ C ∞ (D),theΦ i are elements of TB and the ϕ i t are curves in B with , i = 1, 2.
This completes the calculation of the warp of the Back face; taking into consideration the orientation of the Back face, the warp is w back =−T (∇ Z μ) ∈ Γ TM TA. Finally, the warp of the Left face is the crux of this example. The Left face is the double tangent vector bundle T 2 A for the manifold A. We therefore apply (5). Taking into account the orientation of the Left face, we have We call (ϕ , 0 B ) the bolt of ϕ;in [7] they are called core-linear sections.
Proof That (ϕ , 0 B ) is a linear section follows immediately from the definition of ϕ . Take a 1 , a 2 ∈ A m . Then and similarly for the scalar multiplication. Suppose now that (η, 0 B ) is a linear section. For a ∈ A m , η(a) has the following outline: To show that ϕ is a vector bundle morphism, take a 1 , a 2 ∈ A m . Then follows immediately from the linearity of η. And of course, similarly for the scalar multiplication. Finally, we easily see that ϕ = η.
The following notation prepares the way for Theorem 2.
Definition 7 With the above notation, write R : A → A for the vector bundle map such that It will take us until the end of the section to show that this definition leads to the usual concept of curvature.

The three ultrawarps
The second and final step in applying the warp theorem to the grid (65)onT 2 A consists of calculating the ultrawarps of the induced grids on the core double vector bundles of T 2 A.
In Table 2 we present the induced grids and the ultrawarps, and we explain the calculations in the remainder of the section.
The ultrawarps for the (B-F) and the (U-D) core double vector bundles (first and third row of Table 2) follow directly from Example 1, taking of course into account the orientation of the core double vector bundles.
What requires some work is the ultrawarp of the induced grid on the (L-R) core double vector bundle (second row of Table 2). Taking into account the orientation of the (L-R) core double vector bundle, this ultrawarp is defined by This grid is of a new type; we encounter it here for the first time. Using Definition 7, we rearrange it as We need a small parenthesis on bolts and warps. It is easily seen that the warp of a horizontal linear section (ξ, X ) and a bolt (ϕ , 0 B ) is In addition, we have the following proposition.
The warp of the first grid of (77)is−∇ [Z ,X ] μ (see Example 1). The warp of the second grid of (77), using (76), is −R(μ). So in total, the warp of the induced grid on the (L-R) core double vector bundle is The warp theorem now gives us that Rearranging, we have the standard formula for curvature in terms of covariant derivatives. The use of covariant derivatives and the use of horizontal lifts are the two usual methods for working globally with connections in vector bundles. The covariant derivative formulation was a natural development from the special cases of surface theory and Riemannian geometry. It is our impression that the use of horizontal lifts emerged from comparison with the connection theory of principal bundles.
As noted on page 7, Kobayashi and Nomizu [11] gave two equivalent global formulations of the concept of connection in a principal bundle P(M, G): as a suitable g-valued 1-form on P and as an invariant horizontal distribution H on P. They concentrated on the first of these, and its local version. It was noticed early on that a connection was flat if and only if the corresponding H was involutive, and that the choice of H was equivalent to a lifting of vector fields on M to invariant vector fields on P: putting these together led to a definition of curvature by precisely the formula (74) above. It was then easy to transfer this principal bundle formulation to vector bundles.
Both Dieudonné [4, 17.20] and Greub et al. [10,VII,§4] obtained (78) by iterating the covariant exterior derivative. What we have done is obtain (78) independently of earlier treatments, as a consequence of the warp theorem; if we start with the concept of a connection ∇, and apply the warp theorem to the grid (65)o nT 2 A, we obtain (78)forR(μ). In other words, we have the following theorem. Theorem 2 Given a connection ∇ in (A, q, M) and the equivalent horizontal lifting Z → Z H as defined in (9), the vector bundle map R : A → A for which the bolt R is given by (74) is the curvature R ∇ (Z , X ) as defined in (78).

The triple tangent bundle T 3 M and the Jacobi identity
In this section we consider the triple tangent bundle T 3 M of a manifold M and construct a grid on it, for which the Jacobi identity emerges as a consequence of the warp theorem. A version of this approach was given by Mackenzie [18]. We present here a clearer and more detailed calculation.
Take E to be T 3 M, the triple tangent bundle. This is a special case of Starting with three vector fields X , Y , and Z , each a section of one of the three copies of TM, one can build a grid on T 3 M as follows; see (79) below.
-The front-back linear double section (T ( X ); X , T (X ); X ). Take the complete lift of X across the Down face, and apply the tangent functor to the linear section ( X , X ). The diagram in (79) shows the entire grid.
We now calculate the three ultrawarps defined by this grid. To do this, we calculate the core morphisms of the three linear double sections, and the warps of the six faces.
First, the core morphisms. These follow in an analogous way as in the example of T 2 A, -The core morphism of (T ( X ); X , T (X ); X ) is ( X , X ).
To calculate the warps of the six faces, we take into consideration the orientation of the faces of a triple vector bundle. For the lower faces, by (5) We now calculate the warps of the upper faces.
Back face. The warp of the Back face, for x ∈ TM, is given by Z (x) . prove it globally we have had to introduce a considerable arsenal of background techniques, and make a number of intricate calculations. This is an instance of the great difference between geometric intuition and rigor.
It is natural to consider whether the horizontal lifting process of Sect. 5 and the complete lifting of Sect. 6 can be unified by a general process of lifting. This seems to us unlikely, since complete and horizontal lifts differ in many respects. The warp theorem applies to both cases and can be regarded as providing as much unification as is possible.
In a future paper we will consider in detail the application of the warp theorem to triple vector bundles with compatible bracket structures.