Warps, grids and curvature in triple vector bundles
 242 Downloads
Abstract
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 \(T^3M\) of a manifold and deduce (as earlier) the Jacobi identity. We further apply the result to the triple vector bundle \(T^2A\) for a vector bundle A using a connection in A to define a grid in \(T^2A\). In this case the curvature emerges from the warp theorem.
Keywords
Triple vector bundles Double vector bundles Connections CurvatureMathematics Subject Classification
53C05 18D05 18D35 55R651 Introduction
1.1 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.
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 \(\overline{a}\in TA\). 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.
We now describe the original motivating example for the concepts of grid and warp.
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 \(X\in \varGamma A\) and \(\xi \in \varGamma _BD\) form a linear section of D if \(\xi \) is a morphism of vector bundles over X.
A grid on D is a pair of linear sections \((\xi , X)\) and \((\eta ,Y)\) as shown in (7).
Definition 2
The warp of the grid (7) consisting of \((\xi , X)\) and \((\eta ,Y)\) is \({{\mathrm{w}}}(\xi ,\eta )\in \varGamma C\).
Note that \({{\mathrm{w}}}(\xi ,\eta )\) changes sign if \(\xi \) and \(\eta \) are interchanged. Our convention gives the positive sign to the counterclockwise composition \(\xi \circ Y\).
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 \(\nabla \) be a connection in A.
Recall the horizontal lifting of vector fields from M to A induced by \(\nabla \). To define a vector field on A it is sufficient to define its effect on pullback functions \(f\circ q\) for \(f\in C^{\infty }(M)\), and on linear functions \(\ell _{\varphi }\) for \(\varphi \in \varGamma A^*\), defined by \(\ell _\varphi (a) = \langle \varphi (q(a)),a\rangle \).
It is straightforward to check that \((Z_1+Z_2)^H= Z_1^H+Z_2^H\) and that \((fZ)^H= (f\circ q)Z^H\).
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 \(f\circ q\), \(f\in C^{\infty }(M)\), gives zero.
Lastly, \(\langle \varphi (m),\nabla _{Z(m)}(\mu )\rangle = ((\nabla _{Z}\mu )(m))^\uparrow (\mu (m))(\ell _\varphi ).\) For more details see [16, §3.4].
Suppose given a sigmamap \(\varSigma : TM\times _M A\rightarrow TA\). For \(Z\in \mathfrak {X}(M)\) define \(Z^{H}\in \mathfrak {X}(A)\) by \(Z^{H}(a) = \varSigma (Z(m),a)\) for \(a\in A_m\). This \(Z^{H}\) is a linear vector field over Z. Thus a sigmamap defines a horizontal lifting process, and thus a connection in terms of covariant derivatives \(\nabla _Z\). These processes can be reversed to show that every connection in terms of covariant derivatives \(\nabla _Z\) defines a sigmamap.
It is thus possible to work with connections in vector bundles by using covariant derivatives, by horizontal lifting processes, or by using sigmamaps for the double vector bundle TA. The \(\nabla \) 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): as a suitable \(\mathfrak {g}\)valued 1form 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 sigmamap \(TM\times _M A\rightarrow TA\), and Besse used the corresponding leftsplit map \(TA\rightarrow A\) 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 \(r: T^*A\rightarrow A^*\) to be the composite of \(R^{1}\) with the projection \(T^*(A^*)\rightarrow A^*\). Similarly, use \(R^{1}\) to transport the vector bundle structure of \(T^*(A^*)\rightarrow A^*\) to \(T^*A\rightarrow A^*\). Then the third diagram in (1) is a double vector bundle. The core of \(T^*A\) can be identified with \(T^*M\); given \(\omega \in T^*_mM\), its image \(\overline{\omega }\in T^*A\) is the pullback of \(\omega \) across \(q: A\rightarrow M\) to \(0^A_m\). The map \(R^{1}\) carries \(\overline{\omega }\in T^*A\) to \(\overline{\omega }\in T^*(A^*)\).
Given a section \(\varphi \in \varGamma A^*\), the 1form \(d\ell _\varphi : A\rightarrow T^*A\) is a linear section over \(\varphi \).
Likewise given \(\mu \in \varGamma A\), we obtain a 1form \(d\ell _\mu \) on \(A^*\). Composing with the map \(R: T^*(A^*)\rightarrow T^*A\), we obtain a linear section of \(T^*A\rightarrow A^*\) over \(\mu \).
1.2 Outline of the paper
In a previous paper [18] one of us used a grid in the triple vector bundle \(T^3M\) 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 \(T^3M\) 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. A brief summary is given at the end of the section.
In Sect. 5 we consider a vector bundle A and the triple vector bundle \(T^2A\). A connection in A induces a grid in \(T^2A\), and we show that the concept of curvature arises from the warp theorem; see Theorem 2.
Section 6 presents the example of \(T^3M\) 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.
2 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.
2.1 Basics on triple vector bundles
The basic structure of triple vector bundles has been given in [9, 15, 18].
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 \(E\rightarrow E_{1,2}\) 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 sigmamap. 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.
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 \(E_{1,2,3}\rightarrow E_{1,2}\), \(E_{2,3}\rightarrow E_2\), \(E_{1,3}\rightarrow E_1\) and \(E_3\rightarrow M\). The restriction of \(E_{1,2,3}\rightarrow E_{1,2}\) to \(E_{12,3}\) goes into \(E_{12}\) and inherits the vector bundle structure of \(E_{1,2,3}\rightarrow E_{1,2}\). 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 (UD) core double vector bundle.
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 (BF) and (LR) 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 \(\mathop {{{\mathrm{+}}}}\limits _{1,2}\), \(\mathop {{{\mathrm{+}}}}\limits _{1,3}\), and \(\mathop {{{\mathrm{+}}}}\limits _{2,3}\), 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 \(0^{E_1}:M\rightarrow E_1\), \(m\mapsto 0^{E_1}_m\), with similar notations for \(E_2\) and \(E_3\).
The zero section of \(E_{1,2}\rightarrow E_1\) is denoted by \(\tilde{0}^{1,2}: E_1\rightarrow E_{1,2}\), \(e_1\mapsto \tilde{0}^{1,2}_{e_1}\). The double zero of \(E_{1,2}\) is denoted by \(\odot ^{1,2}_m\), with similar notations for the other vector bundle structures.
The zero section of \(E\rightarrow E_{1,2}\) is denoted by \(\hat{0}:E_{1,2}\rightarrow E\), \(e_{1,2}\mapsto \hat{0}_{e_{1,2}}\). 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 on \(\hat{0}\).
Finally, the triple zero of E is denoted by \(\odot ^3_m\). 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, \(q^{1,2}_2: E_{1,2}\rightarrow E_2\). We omit the corresponding script when the domain is E, for example \(q_{1,2}: E\rightarrow E_{1,2}\), or the target is M, for example \(q^2: E_2\rightarrow M\).
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.
2.2 Result of LiBland and Ševera for cubes of vector bundles
In [13] LiBland and Ševera proved that the condition in the definition of a double vector bundle D on page 2, that the double source map \(\natural : D\rightarrow A\times _M B\) 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 \(A\times _MB\) is a manifold.
Lemma 1
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.
Proof
Proposition 1
If E is a cube of vector bundles as in (11), then the triple source map \(\widetilde{\natural }: E\rightarrow W\) is a surjective submersion.
Proof
Map (20) is surjective since by hypothesis \(\natural ^{\mathsf {B}}\) is a surjective submersion (as the Back face is a double vector bundle). Therefore, the first component of (20) is surjective. Consequently, for any \((e_{1,2},e_{2,3}, e_{1,3})\in W\), there exists an \(e\in E\Big _{e_{2,3} }\) such that \(q_{1,2}(e) = e_{1,2}\), and \(q_{1,3}(e) = e_{1,3}\).
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 LiBland and Ševera on which Proposition 1 is based was established for \({\mathscr {V}}\!{\mathscr {B}}\)Lie groupoids rather than double vector bundles. By combining the techniques of their proof and that above, we could establish a similar result for \({\mathscr {V}}\!{\mathscr {B}}\)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 \(\widetilde{\varSigma }: W \rightarrow E\) that is rightinverse to the triple source map \(E\rightarrow W\).
It seems likely that the existence of sigmamaps 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
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 \(Z_{12}: E_{12}\rightarrow E_{12,3}\). It is a linear section over \(Z: M\rightarrow E_3\).
In a similar fashion we define rightleft and frontback 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 sigmamaps; 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.
2.3 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].
Theorem 1
To give an intrinsic proof, we need to describe the ultrawarps in an alternative way.
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).
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 righthand 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.
3 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 \(m\in M\), 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 \(\mathop {{{\mathrm{\!\!\!\!\!\!}}}}\limits _{1,2}\), \(\mathop {{{\mathrm{\!\!\!\!\!\!}}}}\limits _{1,3}\), and \(\mathop {{{\mathrm{\!\!\!\!\!\!}}}}\limits _{2,3}\).
3.1 First case: two elements that have the same outline
Step 4. Apply the same procedure to Left and Up faces of e and \(e'\).
Step 5. Show that \(u_1 = u_2 = u_3\).
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 \(e'\).
Proposition 2
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.
Example 3
3.2 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.
3.3 Special case: differences between zero elements
3.4 Summary of Sect. 3
 Given two elements \(e,e'\in E\) with the same outline, then they differ by a unique ultracore element \(u\in E_{123}\):$$\begin{aligned} e\mathop {{{\mathrm{\!\!\!\!\!\!}}}}\limits e'\,\triangleright \, u.\end{aligned}$$
 Given two elements \(e,e'\in E\) with two lower faces in common, then their differencedefines a k in one of the core double vector bundles:$$\begin{aligned} e\mathop {{{\mathrm{\!\!\!\!\!\!}}}}\limits e'\,\triangleright \, k,\end{aligned}$$

if \(e,e'\) have the same Right and Down face, then \(k\in E_{23,1}\),

if \(e,e'\) have the same Front and Down face, then \(k\in E_{13,2}\),

if \(e,e'\) have the same Front and Right face, then \(k\in E_{12,3}\).

4 Proof of the warp theorem
4.1 Notation
4.2 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.
Here we write \(\hat{0}_{w_{12}}\mathop {{{\mathrm{+}}}}\limits _{1,3/2,3}u_1\) to denote \(\hat{0}_{w_{12}}\mathop {{{\mathrm{+}}}}\limits _{1,3}u_1=\hat{0}_{w_{12}}\mathop {{{\mathrm{+}}}}\limits _{2,3}u_1\).
4.3 Proof of the warp theorem
We will show that \(u_1+u_2+u_3 = \odot ^3_m\) by showing that \(u_1 = u_2u_3\). There are four steps.
This completes the proof of the warp theorem.
4.4 Summary of Sect. 4
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 noncommutativity 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 noncommutativity 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 (UD) core double vector bundle, which we denoted by \(\lambda _3\). Similarly for the other upper faces, the noncommutativity of the corresponding routes defines \(\lambda _1\) and \(\lambda _2\). The three \(\lambda \)’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 \(\lambda _i\)’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 noncommutativity of four routes, instead of two.
This can be done for the three pairs of \(\lambda _i\) and \(k_i\), and so we obtain the three ultrawarps.
So what does the warp theorem tell us?
Each ultrawarp measures the noncommutativity 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].
5 The triple vector bundle \(T^2A\) and connections in A
In Example 1 of the Introduction we started with a vector bundle \(A\rightarrow M\) and a connection in A, and we saw that the warp of the grid (9) is \(\nabla _Z\mu \), the covariant derivative of \(\mu \) with respect to Z. In this section, starting with a vector bundle \(A\rightarrow M\) and a connection in A, we will build a grid on the triple vector bundle \(T^2A\). 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).
In the first subsection we describe in detail the core double vector bundles of \(T^2A\). In the second subsection we investigate \(J_A~:~T^2A~\rightarrow ~T^2A\), the canonical involution of the manifold A, as a morphism of triple vector bundles. In the third subsection we build the aforementioned grid on \(T^2A\), and in the two remaining subsections we calculate the three ultrawarps induced by this grid.
5.1 The core double vector bundles of \(T^2A\)
An element \(\xi \in TA\) determines core elements of the Back, Left and Up faces. We denote these by \(\overline{\xi }^{B}\), \(\overline{\xi }^{L}\) and \(\overline{\xi }^{U}\), respectively. Since they are elements of T(TA), they can be represented as tangent vectors to curves in TA.
5.2 The canonical involution on \(T^2A\)
The canonical involution \(J_A:T^2A\rightarrow T^2A\) for the manifold A is an isomorphism from the double vector bundle \(T^2A\) to its flip. In what follows we will need to use it as a map of triple vector bundles.
Proposition 3
The map \(J_A\) is an isomorphism of the triple vector bundles shown in (61).
The proof of Proposition 3 relies on the following two lemmas. The first is the naturality property of the canonical involution.
Lemma 2
Let M and N be smooth manifolds, and \(F: M\rightarrow N\) a smooth map. Then \(T^2(F)\circ J_M = J_N\circ T^2(F)\), where \(T^2(F) = T(T(F))\) is the tangent of the tangent map T(F).
Lemma 3
Consider now the maps which \(J_A\) induces on the cores.
5.3 Grids on \(T^2A\)
Now consider a connection \(\nabla \) in A. Example 1 gave a construction of a grid in TA for which the warp is \(\nabla _X\mu \). We now extend this idea to define a grid in \(T^2A\).

From Front to Back face: \((T(X^{H}); X^{H}, T(X);X)\).

From Right to Left face: \((T^2(\mu );T(\mu ),T(\mu );\mu )\).

From Down to Up face: \((\widetilde{Z^{H}}^A;\widetilde{Z},Z^{H};Z)\).

For \((T(X^{H}); X^{H}, T(X);X)\) the core morphism is \((X^{H}, X)\).

For \((T^2(\mu );T(\mu ),T(\mu );\mu )\) the core morphism is \((T(\mu ), \mu )\).

For \((\widetilde{Z^{H}}^A;\widetilde{Z},Z^{H};Z)\) the core morphism is \((Z^{H},Z)\).
This completes the proof that the core morphism of \((\widetilde{Z^{H}}^A;\widetilde{Z},Z^{H};Z)\) is \((Z^{H},Z)\).
5.4 The six warps of grid (65)
Warps of grids of each face of \(T^2A\)
\({{\mathrm{w}}}_{\text {back}}\)  \({{\mathrm{w}}}_{\text {front}}\) 

\(\widetilde{Z^{H}}^A\circ T(\mu ) T^2(\mu )\circ \widetilde{Z} \,\triangleright \,T(\nabla _Z\mu )\)  \(Z^H\circ \mu T(\mu )\circ Z \,\triangleright \, \nabla _Z\mu \) 
\({{\mathrm{w}}}_{\text {left}}\)  \({{\mathrm{w}}}_{\text {right}}\) 

\(T(X^{H})\circ Z^{H}  \widetilde{Z^{H}}^A\circ X^{H}\,\triangleright \, [Z^{H},X^{H}]\)  \(T(X)\circ Z  \widetilde{Z}\circ X\,\triangleright \, [Z,X]\) 
\({{\mathrm{w}}}_{\text {up}}\)  \({{\mathrm{w}}}_{\text {down}}\) 

\(T^2(\mu )\circ T(X)  T(X^{H})\circ T(\mu )\,\triangleright \, T(\nabla _X\mu )\)  \(T(\mu )\circ X  X^H\circ \mu \,\triangleright \, \nabla _X\mu \) 
The three warps \({{\mathrm{w}}}_{\text {front}}\), \({{\mathrm{w}}}_{\text {right}}\), and \({{\mathrm{w}}}_{\text {down}}\) 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.
Proposition 4
Let \((\xi ,X)\) and \((\eta ,Y)\) be a grid on a double vector bundle D with warp \({{\mathrm{w}}}(\xi ,\eta )\in \varGamma C\). Then \((T(\xi ),T(X))\) and \((T(\eta ),T(Y))\) form a grid on the double vector bundle TD in (67) below and the warp of the tangent grid \((T(\xi ),T(X))\), \((T(\eta ),T(Y))\) is \(T({{\mathrm{w}}}(\xi ,\eta ))\in \varGamma _{TM}(TC)\).
Proof
We leave the verification that TD is a double vector bundle and that \((T(\xi ),T(X))\) and \((T(\eta ),T(Y))\) are linear sections of TD to the reader.
Using the warp theorem, we now show that the warp \([Z^{H}, X^{H}]\), combined with the other warp terms from the grid (65), yields the standard formula for curvature in terms of covariant derivatives. For the full statement see Theorem 2.
To begin, note that both \([Z^{H},X^{H}]\) and \([Z,X]^{H}\) project to [Z, X] and therefore their difference is a linear and vertical vector field on A.
At this point we take a closer look at linear sections \((\eta , 0^B)\), where \(\eta \in \varGamma _A D\) and \(0^B\in \varGamma B\), of a general double vector bundle D.
Lemma 4
We call Open image in new window the bolt of\(\varphi \); in [7] they are called corelinear sections.
Proof
The following notation prepares the way for Theorem 2.
Definition 7
It will take us until the end of the section to show that this definition leads to the usual concept of curvature.
5.5 The three ultrawarps
The second and final step in applying the warp theorem to the grid (65) on \(T^2A\) consists of calculating the ultrawarps of the induced grids on the core double vector bundles of \(T^2A\).
Induced grids and ultrawarps in \(T^2A\)
The ultrawarps for the (BF) and the (UD) 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.
Proposition 5
We leave the proof to the reader.
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 \(\mathfrak {g}\)valued 1form on P and as an invariant horizontal distribution \(\mathscr {H}\) 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 \(\mathscr {H}\) was involutive, and that the choice of \(\mathscr {H}\) 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 \(\nabla \), and apply the warp theorem to the grid (65) on \(T^2A\), we obtain (78) for \(R(\mu )\). In other words, we have the following theorem.
Theorem 2
Given a connection \(\nabla \) in (A, q, M) and the equivalent horizontal lifting \(Z\mapsto Z^H\) as defined in (9), the vector bundle map \(R : A\rightarrow A\) for which the bolt Open image in new window is given by (74) is the curvature \(R_\nabla (Z,X)\) as defined in (78).
6 The triple tangent bundle \(T^3M\) and the Jacobi identity
In this section we consider the triple tangent bundle \(T^3M\) 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.

The frontback linear double section \((T(\widetilde{X});\widetilde{X},T(X);X)\). Take the complete lift of X across the Down face, and apply the tangent functor to the linear section \((\widetilde{X},X)\).

The rightleft linear double section \((T^2(Y);T(Y),T(Y);Y)\). Apply the tangent functor to Y and then to T(Y).

The downup linear double section \((\widetilde{\widetilde{Z}};\widetilde{Z},\widetilde{Z};Z)\). Take the complete lift of Z across the Front face, and the complete lift of this across the Left face. Likewise take the complete lift of Z across the Right face. One does need to check that \((\widetilde{\widetilde{Z}},\widetilde{Z})\) is indeed a linear section of the Back face.

The core morphism of \((T(\widetilde{X});\widetilde{X},T(X);X)\) is \((\widetilde{X},X)\).

The core morphism of \((T^2(Y);T(Y),T(Y);Y)\) is (T(Y), Y).

The core morphism of \((\widetilde{\widetilde{Z}};\widetilde{Z},\widetilde{Z};Z)\) is \((\widetilde{Z},Z)\).

For the Front face: \(\widetilde{Z}(Y){{\mathrm{\!\!\!\!\!\!}}}T(Y)(Z)\,\triangleright \,[Y,Z]\).

For the Right face: \(T(X)(Z){{\mathrm{\!\!\!\!\!\!}}}\widetilde{Z}(X)\,\triangleright \,[Z,X]\).

For the Down face: \(T(Y)(X){{\mathrm{\!\!\!\!\!\!}}}\widetilde{X}(Y)\,\triangleright \,[X,Y]\).
Up face. For the Up face, using Proposition 4, \({{\mathrm{w}}}_{\text {up}} = T([X,Y])\) follows directly.
The three ultrawarps. The three core double vector bundles are all copies of \(T^2M\), and their ultracore is \(TM\rightarrow M\).
7 Concluding remarks
The statement of the warp theorem is easy to describe and needs only minimal notation (in fact it reminds us of the “boundary of the boundary is zero” for cochains), but to 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.
Footnotes
 1.
We are grateful to Rajan Mehta for pointing this reference out to us.
Notes
Acknowledgements
We are very grateful to Yvette Kosmann–Schwarzbach, Ted Voronov and Ping Xu for valuable comments at various stages in the preparation of this paper. We also send our best thanks to Fani Petalidou for thorough readings of earlier versions of the paper and for catching many slips. We especially appreciate the comments of the referee, which have clarified and enhanced our own understanding of the results.
References
 1.Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications, Applied Mathematical Sciences, vol. 75, 2nd edn. Springer, New York (1988)CrossRefzbMATHGoogle Scholar
 2.Besse, A.L.: Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93. Springer, Berlin (1978). With appendices by D. B. A. Epstein, J.P. Bourguignon, L. BérardBergery, M. Berger and J. L. KazdanGoogle Scholar
 3.Courant, T.: Tangent Dirac structures. J. Phys. A 23(22), 5153–5168 (1990). http://stacks.iop.org/03054470/23/5153
 4.Dieudonné, J.: Treatise on analysis. Vol. III. Academic Press, New York (1972). Translated from the French by I. G. MacDonald, Pure and Applied Mathematics, Vol. 10IIIGoogle Scholar
 5.Flari, M. K.: Triple vector bundles in Differential Geometry, PhD thesis, University of Sheffield (2018)Google Scholar
 6.Grabowski, J., Rotkiewicz, M.: Higher vector bundles and multigraded symplectic manifolds. J. Geom. Phys. 59(9), 1285–1305 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
 7.GraciaSaz, A., Jotz Lean, M., Mackenzie, K.C.H., Mehta, R.A.: Double Lie algebroids and representations up to homotopy. J. Homotopy Relat. Struct. 13(2), 287–319 (2018). https://doi.org/10.1007/s4006201701831 MathSciNetCrossRefzbMATHGoogle Scholar
 8.GraciaSaz, A., Mackenzie, K.C.H.: Duality functors for \(n\)fold vector bundles. arXiv:1209.0027
 9.GraciaSaz, A., Mackenzie, K.C.H.: Duality functors for triple vector bundles. Lett. Math. Phys. 90(1–3), 175–200 (2009). https://doi.org/10.1007/s110050090346z ADSMathSciNetCrossRefzbMATHGoogle Scholar
 10.Greub, W., Halperin, S., Vanstone, R.: Connections, Curvature, and Cohomology, Vol. II: Lie Groups, Principal Bundles, and Characteristic Classes. Academic Press, New York (1973)zbMATHGoogle Scholar
 11.Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Vol. I Interscience Publishers. Wiley, New York (1963)Google Scholar
 12.Koszul, J.L.: Lectures on fibre bundles and differential geometry. Notes by S. Ramanan. Tata Institute of Fundamental Research Lectures on Mathematics, No. 20. Tata Institute of Fundamental Research, Bombay (1965)Google Scholar
 13.LiBland, D., Ševera, P.: QuasiHamiltonian groupoids and multiplicative Manin pairs. Int. Math. Res. Not. IMRN 10, 2295–2350 (2011). https://doi.org/10.1093/imrn/rnq170 MathSciNetzbMATHGoogle Scholar
 14.Mackenzie, K.C.H.: Double Lie algebroids and secondorder geometry. I. Adv. Math. 94(2), 180–239 (1992). https://doi.org/10.1016/00018708(92)90036K MathSciNetCrossRefzbMATHGoogle Scholar
 15.Mackenzie, K.C.H.: Duality and triple structures. In: The breadth of symplectic and Poisson geometry, Progr. Math., vol. 232, pp. 455–481. Birkhäuser, Boston, MA (2005)Google Scholar
 16.Mackenzie, K.C.H.: General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, vol. 213. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
 17.Mackenzie, K.C.H.: Ehresmann doubles and Drinfel’d doubles for Lie algebroids and Lie bialgebroids. J. Reine Angew. Math. 658, 193–245 (2011)MathSciNetzbMATHGoogle Scholar
 18.Mackenzie, K.: Proving the Jacobi identity the hard way. In: Geometric methods in physics, Trends Math., pp. 357–366. Birkhäuser, Basel (2013)Google Scholar
 19.Mackenzie, K.C.H., Xu, P.: Lie bialgebroids and Poisson groupoids. Duke Math. J. 73(2), 415–452 (1994). https://doi.org/10.1215/S0012709494073183 MathSciNetCrossRefzbMATHGoogle Scholar
 20.Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W. H. Freeman and Co., San Francisco (1973)Google Scholar
 21.Pradines, J.: Fibrés vectoriels doubles et calcul des jets non holonomes, Esquisses Mathématiques [Mathematical Sketches], vol. 29. Université d’Amiens, U.E.R. de Mathématiques, Amiens (1977)Google Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.