polysymplectic and multisymplectic structures on manifolds ...forger/pdffiles/teselgg.pdf · the...

Polysymplectic and Multisymplectic Structures

on Manifolds and Fiber Bundles

Leandro Gustavo Gomes

THESIS SUBMITTEDTO THE

INSTITUTE FOR MATHEMATICS AND STATISTICSOF THE

UNIVERSITY OF SAO PAULOIN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR OBTAINING THE DEGREE OFDOCTOR OF PHILOSOPHY

Area : Applied MathematicsSupervisor : Prof. Dr. Frank Michael Forger

Financially supported by CNPq and CAPES

- Sao Paulo, February 2007 -

Abstract

In this thesis, we introduce a new class of multilinear alternating forms and of differentialforms called polylagrangian (in the case of vector-valued forms) or multilagrangian (in thecsae of forms that are partially horizontal with respect to a given subspace or subbundle),characterized by the existence of a special type of maximal isotropic subspace or subbundlecalled polylagrangian or multilagrangian, respectively. As it turns out, these constitute theadequate framework for the formulation of an algebraic Darboux theorem. Combining thisnew algebraic structure with standard integrability conditions (dω = 0) allows us to derivea geometric Darboux theorem (existence of canonical local coordinates). Polysymplecticand multisymplectic structures, including all those that appear in the covariant hamiltonianformalism of classical field theory, are contained as special cases.

Contents

Introduction i

1 Polylagrangian and multilagrangian forms: algebraic theory 1

1.1 Vector-valued forms on linear spaces . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Partially horizontal forms on linear spaces . . . . . . . . . . . . . . . . . . . 13

1.3 The symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Canonical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4.1 The canonical polylagrangian form . . . . . . . . . . . . . . . . . . . 19

1.4.2 The canonical multilagrangian form . . . . . . . . . . . . . . . . . . . 20

1.5 Polylagrangian vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.6 Multilagrangian vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Polylagrangian and multilagrangian forms: differential theory 29

2.1 Cartan calculus for vertical forms . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2 Polylagrangian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3 Poly- and multilagrangian fiber bundles . . . . . . . . . . . . . . . . . . . . . 35

2.4 The symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5 The Darboux Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1

2 Contents

3 Poly- and multisymplectic structures 45

3.1 Polysymplectic vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Multisymplectic vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3 Poly- and multisymplectic manifolds and fiber bundles . . . . . . . . . . . . 59

A Poincare Lemma 61

Introduction

Multisymplectic geometry is increasingly recognized as providing the appropriate mathe-matical framework for classical field theory from the hamiltonian point of view – just assymplectic geometry does for classical mechanics. Unfortunately, the development of thisnew area of differential geometry has so far been hampered by the non-existence of a fullysatisfactory definition of the concept of a multisymplectic structure. Ideally, such a definitionshould be mathematically simple as well as in harmony with the needs of applications tophysics.

The present thesis is devoted to presenting a new proposal for overcoming this problem.

As a first step, let us consider a simple analogy. The symplectic forms encountered inclassical mechanics can locally all be written in the form

ω = dqi ∧ dpi , (1)

where q1, . . . , qN , p1, . . . , pN are a particular kind of local coordinates on phase space knownas canonical coordinates or Darboux coordinates. Introducing time t and energy E as addi-tional variables (which is essential, e.g., for incorporating non-autonomous systems into thesymplectic framework of hamiltonian mechanics), this equation is replaced by

ω = dqi ∧ dpi + dE ∧ dt , (2)

where t, q1, . . . , qN , p1, . . . , pN , E can be viewed as canonical coordinates on an extendedphase space. Similarly, the multisymplectic forms encountered in classical field theory overan n-dimensional space-time manifold M can locally all be written in the form

ω = dqi ∧ dpµi ∧ dnxµ − dp ∧ dnx , (3)

where xµ, qi, pµi , p (1 6 µ 6 n, 1 6 i 6 N) can again be viewed as canonical coordinates on

some extended multiphase space. Here, the xµ are (local) coordinates for M , while p is still

i

ii Introduction

a single energy variable (except for a sign), dnx is the (local) volume form induced by thexµ and dnxµ is the (local) (n−1)-form obtained by contracting dnx with ∂µ ≡ ∂/∂xµ:

dnxµ = i∂µdnx .

The idea of introducing “multimomentum variables” labeled by an additional space-timeindex µ (n multimomentum variables p µ

i for each position variable qi) goes back to thework of de Donder [5] and Weyl [20] in the 1930’s (and perhaps even further) and has beenrecognized ever since as being an essential and unavoidable ingredient in any approach toa generally covariant hamiltonian formulation of classical field theory. Understanding theproper geometric setting for this kind of structure, however, has baffled both mathematiciansand physicists for decades, as witnessed by the many attempts that can be found in theliterature, almost all of which have ultimately failed, at some point or other, either becausethey did not properly take into account one or several of the various subtleties involved inthe problem or else because they took “the easy way out” by artificially invoking additionalstructures. A typical example of such an additional structure is the choice of a backgroundconnection, which excludes the possibility to incorporate gauge theories where connectionsare part of the dynamical variables: obviously, a formalism that claims to be general butfails to accommodate gauge theories can hardly be of much use.

A notable exception is provided by the cojet bundle construction of multiphase space infirst order classical field theories whose hamiltonian formulation is obtained from a lagrangianformulation via a Legendre transformation. Starting out from a fiber bundle E over space-time M whose sections are regarded as the basic fields of the model under consideration, oneforms its first order jet bundle JE which is a fiber bundle over M (with respect to the sourceprojection) whose sections represent the possible first order derivatives of the basic fields andwhich is the natural arena for the lagrangian formalism. Next, one makes use of the fact thatJE is also an affine bundle over E (with respect to the target projection) to form its affinedual P = J©?E which turns out to be the natural arena for the hamiltonian formalism; inparticular, it carries a globally defined multisymplectic form ω whose local representationin canonical coordinates has the form given in equation (3) above. Remarkably, in this case,the multisymplectic form ω is (up to a sign) the exterior derivative of a globally definedmulticanonical form θ whose local representation in canonical coordinates reads

θ = pµi dqi ∧ dnxµ + p dnx . (4)

We shall refrain from describing this construction in more detail, even though it providesthe only known class of examples of multisymplectic structures known to date, since it isextensively documented in the literature [4, 6, 8, 9]. However, it should be noted that it isstrictly analogous to the cotangent bundle construction of phase space in mechanics, whichclearly indicates that it should not be the most general such construction. What is lackingis the multisymplectic analogue of other examples of symplectic manifolds which cannot be

Introduction iii

represented as vector bundles and where ω fails to be exact, such as Kahler manifolds orcoadjoint orbits.

As a first attempt to approach the problem, consider the obvious option of extendingthe concept of a symplectic structure to that of a multisymplectic structure by defining amultisymplectic manifold to be a manifold P equipped with a differential form ω of degreen + 1 which is closed (dω = 0) as well as everywhere non-degenerate, i.e., satisfies

iwω(p) = 0 =⇒ w = 0 for p ∈P , w ∈TpP . (5)

Obviously, this definition captures part of the relevant aspects but is much too broad to beof any practical use: in particular, it is insufficient to guarantee the validity of a Darbouxtype theorem (except when n = 1, of course).

The main problem with the definition just given is that it completely ignores the roleof space-time. More precisely, it ignores the fact that the manifold P is not simply just amanifold but rather the total space of a fiber bundle over a base manifold M which not onlydetermines the degree of the form in question (dim M = n ⇒ deg ω = n + 1) but alsoprovides a notion of horizontal forms (the dxµ and their exterior products) which, roughlyspeaking, allows us to split off an (n−1)-horizontal factor from ω and thus reduce it to abunch of 2-forms, according to

ω = ω(µ) ∧ dnxµ , (6)

whereω(µ) = dqi ∧ dpµ

i − dp ∧ dxµ , (7)

sincedxµ ∧ dnxν = δµ

ν dnx . (8)

Of course, such a splitting is coordinate dependent, but its existence is an important ingre-dient which must not be neglected but rather be reformulated in a coordinate independentmanner.

In essence, this means that there is no natural notion of a multisymplectic manifoldbut only that of a multisymplectic fiber bundle, whose base space is to be interpreted asthe space-time manifold on which classical fields are defined.1 For our purposes here, thebase space need not carry any additional structure, such as a Lorentz metric or, in theEuclidean setting, a Riemann metric, but it must be present as a carrier space. This maysound strange when compared with the situation in mechanics, where one is used to dealingwith symplectic manifolds, rather than symplectic fiber bundles, but that is really a specificfeature of non-relativistic mechanics for autonomous systems which gets lost when one passesto non-autonomous systems or to relativistic mechanics:

1Indeed, the most general mathematical concept of a field, in classical field theory, is that fields aresections of fiber bundles over space-time.

iv Introduction

• Even in the realm of non-relativistic classical mechanics, the treatment of non-autonomous systems requires extending the conventional phase space of the theory,which is a symplectic manifold P0, say, by incorporating a copy of the real line R asa “time axis” and, if one wants to avoid working with contact manifolds such as the“simply extended phase space” R× P0, incorporating another copy of the real line Ras an “energy axis”: this allows us to remain within the realm of symplectic manifolds,specifically the “doubly extended phase space” R × P0 × R, already considered byE. Cartan. Both of these extensions are, in a very obvious way, fiber bundles over thetime axis R, and it is this structure that will be reproduced naturally when our defi-nition of a multisymplectic fiber bundle is specialized to the case of a one-dimensionalbase manifold.

• In relativistic classical mechanics, the notion of time suffers a radical change of inter-pretation when compared to the non-relativistic case. There, what prevails is Newton’snotion of absolute time, which can be and is used as a universal parameter for thesolutions of ordinary differential equations, for all possible dynamical systems at once.In the transition to the relativistic case, time loses its distinguished status and becomesa coordinate (almost) like any other one – after all, temporal and spatial coordinatesmix under Lorentz transformations. Therefore, it is absolutely natural to combineit with the generalized positions and momenta of mechanics since it is no longer theindependent variable but rather one of the dependent variables, whereas the new in-dependent variable is something different: for particle trajectories, it is proper time asmeasured by a clock moving along with the particle along its trajectory, rather thancoordinate time as measured in the reference frame of a distant observer.

In short, mechanics does provide a natural setting for the appearance of symplectic fiberbundles (over the real line).

Taking into account these considerations, we can make a second attempt at defining theconcept of a multisymplectic structure, according to which a multisymplectic fiber bundleis a fiber bundle P over an n-dimensional manifold M equipped with a differential form ωof degree n + 1 which is closed (dω = 0) as well as everywhere non-degenerate and (n−1)-horizontal, i.e., satisfies equation (5) as before and

iv1iv2

iv3ω(p) = 0 for p ∈P , v1, v2, v3 ∈VpP , (9)

where V P denotes the vertical bundle (V P = ker Tπ where Tπ : TP → TM is the tangentmap to the bundle projection π : P →M). As we shall see, these conditions allow us to

• Introduce a “joint vertical kernel” of ω, which is an involutive vector subbundle KωPof V P defined by

(Kω)pP = v ∈VpP | iviv′ω(p) = 0 for all v′ ∈VpP , (10)

Introduction v

and which turns out to be at most one-dimensional: it can either be trivial, as in thecase of the “simply extended phase space” of mechanics, or else be one-dimensional, asin the “doubly extended phase space” of mechanics, where it represents the additionalenergy variable.

• Reduce the original “multisymplectic” form ω to a “polysymplectic” form ω on the ver-tical bundle (i.e., a family of “polysymplectic” forms ωx on the fibers Px parametrizedby the points x in the base space M), which is formally a homomorphism

ω :∧2

V P −→ π∗(∧n−1

T ∗M)

(11)

of vector bundles over P , for which KωP is the “joint kernel”.

Unfortunately, these conditions are still not sufficient to guarantee the (local) existence ofcanonical coordinates. In fact, simple counterexamples show that something important ismissing, namely a further algebraic condition that allows us to find a Darboux basis in thetangent space at each point. A second problem is then to figure out whether (or under whatassumptions) it is possible to even find a holonomic Darboux basis in a neighborhood of eachpoint.

As will be shown in the present thesis, both questions have simple and elegant answers,thus giving rise to a new and, in our view, finally adequate definition of the notion of amultisymplectic structure. Hopefully, this conceptual clarification will open the way to newmathematical insights that promise to be highly relevant for physics.

The main body of this thesis is divided into two chapters: the first is purely algebraic whilethe second has a geometric flavor. At the algebraic level, the new mathematical structurethat has emerged in the course of our investigation and has turned out to be remarkablygeneral since it appears in a wide variety of different situations, can be resumed as therequirement of “existence of a special type of maximal isotropic subspace” which we shallcall a “polylagrangian subspace” or “multilagrangian subspace”, depending on the context.This concept allows to handle more general forms than the standard polysymplectic or multi-symplectic forms and we propose to call them “polylagrangian” or “multilagrangian” forms,respectively, since they turn out to constitute the adequate framework for formulating an al-gebraic Darboux theorem which guarantees the existence of a canonical basis. Subsequently,when passing from the realm of (multi)linear algebra to that of differential geometry, this newstructure reveals its full power since combining it with the standard integrability conditionfor differential forms (dω = 0), we obtain a geometric Darboux theorem which guaranteesthe existence of canonical local coordinates. In the third and final chapter, we specify thegeneral results obtained before to polysymplectic and multisymplectic forms and also discussthe relation between our approach and previous attempts of other authors.

Chapter 1

Polylagrangian and multilagrangianforms: algebraic theory

In this chapter we investigate two specific classes of alternating multilinear forms on vectorspaces that play an important role in classical field theory and which we shall call poly-lagrangian forms (containing polysymplectic forms as a special case) and multilagrangianforms (containing multisymplectic forms as a special case). The former are vector-valuedwhile the latter are partially horizontal, i.e., restricted to satisfy a certain degree of horizon-tality relative to some fixed “vertical” subspace. We begin by defining a few general notions,such as that of the kernel, the support and the rank of an alternating multilinear form, andthen introduce various types of orthogonal complements that will be used in the definitionof certain classes of subspaces which generalize the isotropic, coisotropic, lagrangian andmaximal isotropic subspaces of symplectic geometry. One of the central points of this thesiswill be to show that there is a specific extension of the concept of a lagrangian subspace forwhich we shall introduce the term “polylagrangian subspace” (in the case of vector-valuedforms) or “multilagrangian subspace” (in the case of partially horizontal forms) and whoseexistence turns out to be the key to developing a non-trivial theory of polysymplectic andmultisymplectic forms covering all situations studied so far: the ones treated in the existingmathematical literature as well as the ones relevant for applications to physics, which havenot yet been adequately treated.

The first section of this chapter will be devoted to some general properties of vector-valued forms. First, recall that given two finite-dimensional 1 vector spaces, V and T , aT -valued k-form on V is a linear mapping

ω :∧k

V −→ T , (1.1)

1The hypothesis that all vector spaces considered in this work should be finite-dimensional is imposed tosimplify the presentation of the ideas but is by no means indispensable: it could be replaced by adequateconditions from functional analysis. This generalization will be left to a possible future investigation.

1

2 Polylagrangian and multilagrangian forms: algebraic theory

or in other words, an alternating k-multilinear mapping from V × . . .× V (k factors) to T .The vector space of such forms will be denoted by

Lka(V ; T ) ∼= L

(∧kV ; T

) ∼= (∧kV ∗)⊗ T . (1.2)

Taking T = R we recover the definition of an ordinary k-form on V .

The second section of this chapter will be devoted to ordinary forms which are partiallyhorizontal. To understand what this means, fix a finite-dimensional 1 vector space W togetherwith a subspace V , called the vertical space, and consider the quotient space T := W/V ,called the base space. Denoting the canonical projection of W onto T by π, we have anexact sequence of vector spaces

0 −→ V −→ Wπ−→ T −→ 0 . (1.3)

In this situation, recall that a k-form α ∈∧k

W ∗ on W is horizontal (relative to π) if itscontraction iv α with any vertical vector v ∈ V = ker π vanishes [10, Vol. 2], [16, Vol. 1].

More generally, we say that a k-form α ∈∧k

W ∗ on W is (k− r)-horizontal (relative to π),where 0 6 r 6 k , if its contraction with more than r vertical vectors vanishes, i.e., if

iv1 . . . ivr+1 α = 0 for v1, . . . , vr+1 ∈V . (1.4)

The vector space of (k− r)-horizontal k-forms on W will be denoted by∧ k

r W ∗. Obviously,

the condition of 0-horizontality (r = k) is void, so∧ k

k W ∗ =∧k

W ∗, while the k-horizontalk-forms (r = 0) are precisely the horizontal forms as defined before, and we have a canonical

isomorphism∧ k

0 W ∗ ∼=∧k

T ∗. For posterior use, we note that in the direction ←, i.e., as alinear map ∧k

T ∗ −→∧ k

0 W ∗

αT 7−→ αW

(1.5)

it is given by pull-back with the projection π, i.e., by αW = π∗(αT ), or explicitly,

αW (w1, . . . , wk) = αT (π(w1), . . . , π(wk)) for w1, . . . , wk ∈W . (1.6)

Furthermore, we have a sequence of inclusions∧kT ∗ ∼=

∧ k0 W ∗ ⊂ . . . ⊂

∧ kr W ∗ ⊂ . . . ⊂

∧ kk W ∗ =

∧kW ∗ , (1.7)

where the first few terms can be trivial, since∧ kr W ∗ = 0 if k − r > dim T . (1.8)

3

This motivates the interpretation of such forms as “partially horizontal”. More pre-cisely, we we can justify this terminology by observing that if we introduce a basiseV

1 , . . . , eVm, eT

1 , . . . , eTn of W such that the first m vectors span V while the last n vec-

tors span a subspace complementary to V and hence isomorphic to T , then in terms of the

dual basis e1V , . . . , em

V , e1T , . . . , en

T of W ∗, an arbitrary form α ∈∧ k

r W ∗ can be representedas

α =r∑

s=0

1s!

1(k−s)!

αi1... is; µ1... µk−sei1

V ∧ . . . ∧ eisV ∧ eµ1

T ∧ . . . ∧ eµk−s

T , (1.9)

showing that forms α ∈∧ k

r W ∗ are (k − r)-horizontal in the sense of being represented aslinear combinations of exterior products of 1-forms among which at least k−r are horizontal,i.e., belong to the subspace V ⊥ (which is spanned by the 1-forms e1

T , . . . , enT ).2 This also

means that ei1V ∧ . . . ∧ eis

V ∧ eµ1

T ∧ . . . ∧ eµk−s

T | 0 6 s 6 r , 1 6 i1 < . . . < is 6 dim V ,

1 6 µ1 < . . . < µk−s 6 dim T is a basis of∧ k

r W ∗ and so we have

dim∧ k

r W ∗ =r∑

s=0

(dim V

s

)(dim T

k − s

), (1.10)

where it is to be understood that, by definition,(

pq

)= 0 if q > p .

In the third section we shall discuss a construction that allows us to associate to eachpartially horizontal form a certain vector-valued form: the latter will be called the “symbol”of the former in order to stress the analogy with the concept of the (leading) symbol of adifferential operator. This procedure provides the link between polysymplectic and multi-symplectic structures. Briefly, in the notation of the previous paragraph, the symbol of

a (k − r)-horizontal k-form α on W , α ∈∧ k

r W ∗, is the∧k−r

T ∗-valued r-form α on V ,

α ∈∧r

V ∗ ⊗∧k−r

T ∗, given by

α(v1, . . . , vr) = iv1. . . ivr

α , for v1, . . . , vr ∈V . (1.11)

where we use the identifications

Lra

(V ;∧k−r

T ∗) ∼= L(∧r

V ;∧k−r

T ∗) ∼= ∧rV ∗ ⊗

∧k−rT ∗ , (1.12)

and ∧k−rT ∗ ∼=

∧k−r0 W ∗ . (1.13)

For a better understanding of this construction, introduce a basis eV1 , . . . , eV

m, eT1 , . . . , eT

nof W with dual basis e1

V , . . . , emV , e1

T , . . . , enT of W ∗, as before; then if the form α ∈

∧ kr W ∗

2The symbol “.⊥” stands for the annihilator, that is, V ⊥ is the subspace consisting of all linear forms inW ∗ which vanish on V .


has the expansion

α =r∑

s=0

1s!

1(k−s)!

αi1... is; µ1... µk−sei1

V ∧ . . . ∧ eisV ∧ eµ1

T ∧ . . . ∧ eµk−s

T , (1.14)

the form α ∈∧r

V ∗ ⊗∧k−r

T ∗ is given by the expansion

α = 1r!

1(k−r)!

αi1... ir; µ1... µk−rei1

V ∧ . . . ∧ eirV ⊗ eµ1

T ∧ . . . ∧ eµk−r

T . (1.15)

Hence passage to the symbol can be regarded as a projection∧ kr W ∗ −→

∧rV ∗ ⊗

∧k−rT ∗

α 7−→ α(1.16)

whose kernel is the subspace∧ k

r−1W∗.

An auxiliary tool often employed in the context described in the last two paragraphs isthe choice of a splitting of the exact sequence (1.3), i.e., a linear mapping s : T → W suchthat π s = idT , or equivalently, the choice of a subspace H of W , called the horizontalsubspace, which is complementary to V , i.e., which satisfies

W = V ⊕H , (1.17)

with H the image of s in W and s the inverse of the restriction of the projection π to H.Then it is easy to see that

W ∗ = H⊥ ⊕ V ⊥ , (1.18)

where H⊥ ∼= V ∗ and V ⊥ ∼= H∗, and more generally,

∧ krW

∗ ∼=r⊕

s=0

∧sH⊥ ⊗

∧k−sV ⊥ . (1.19)

Returning to the canonical isomorphism∧ k

0 W ∗ ∼=∧k

T ∗, we note that in the direction →,i.e., as a linear map ∧ k

0 W ∗ −→∧k

T ∗

αW 7−→ αT

(1.20)

it is explicitly defined as the pull-back by s, i.e., αT = s∗(αW ) or

αT (t1, . . . , tk) = αW (s(t1), . . . , s(tk)) for t1, . . . , tk ∈T . (1.21)

1.1 Vector-valued forms on linear spaces 5

Thus we arrive at the following explicit formula for the symbol α ∈∧r

V ∗ ⊗∧k−r

T ∗ of a

partially horizontal form α ∈∧ k

r W ∗ :

α(v1, . . . , vr)(t1, . . . , tk−r

)= α(v1, . . . , vr, s(t1), . . . , s(tk−r))

for v1, . . . , vr ∈V , t1, . . . , tk−r ∈T .(1.22)

However, it is worthwhile to emphasize that the last two formulas, like all the constructionsdescribed in the last two paragraphs, do not depend on the choice of the splitting s.

In the fourth section we present the construction of canonical models of polylagrangianand multilagrangian forms. Finally, in the last two sections, we show that general poly-lagrangian and multilagrangian forms can always be brought into this canonical form, bymeans of an appropriate isomorphism. When expressed in terms of adapted bases, thisstatement can be considered an “algebraic precursor” of the Darboux theorem.

In what follows we shall adopt the general convention that objects related to vector-valued forms will be characterized by a hat, just as above. Moreover, since we are interestedonly in forms of degree k > 1, we shall relabel the parameter k and consider ordinary (k+1)-

forms, ω ∈∧k+1

W ∗ , as well as vector-valued (k + 1)-forms, ω ∈∧k+1

V ∗ ⊗ T , with k > 1.Finally, the dimensions of the “auxiliary” spaces T and T will be denoted by n and by n,respectively.

The results obtained in this chapter are all formulated for vector spaces over the realnumbers, but they remain valid in the context of vector spaces over arbitrary fields of char-acteristic zero.

1.1 Vector-valued forms on linear spaces

In this section, we shall first present a few general notions from the theory of multilinearforms that can be easily extended from ordinary forms to the more general context of vector-valued forms. As a major new concept, we then introduce a special type of maximal isotropicsubspace, called polylagrangian subspace, which plays a fundamental role in all that follows.

Let V and T be finite-dimensional vector spaces, with dim T = n, and let ω be a T -valued(k + 1)-form on V :

ω ∈∧k+1

V ∗ ⊗ T . (1.23)

The kernel of ω is the subspace ker ω of V defined by

ker ω = v ∈V | ivω = 0 , (1.24)

that is,ker ω = v ∈V | ω(v, v1, . . . , vk) = 0 for all v1, . . . , vk ∈V . (1.25)


When ker ω = 0 we say that ω is non-degenerate. The support of ω is the subspacesupp ω of V ∗ defined as the annihilator of ker ω, that is,3

supp ω = (ker ω)⊥ = v∗ ∈V ∗ | 〈v∗, v〉 = 0 for all v ∈ ker ω . (1.26)

Conversely, we have

ker ω = (supp ω)⊥ = v ∈V | 〈v∗, v〉 = 0 for all v∗ ∈ supp ω . (1.27)

According to a frequently employed convention (see, for instance, Ref. [18]), the rank rk(ω)of ω is defined to be the dimension of its support:

rk(ω) = dim supp ω . (1.28)

However, depending on the specific nature of ω, we shall often find it more convenient toadopt a numerically different definition. For example, the rank of a symplectic form on a2N -dimensional linear space is usually defined to be N , rather than 2N . The same kind ofphenomenon will appear for the various types of forms to be introduced below.

One useful way of thinking about vector-valued forms is to consider them as “multiplets”of ordinary forms, relative to some basis of the auxiliary space. More specifically, with eachelement t∗ ∈ T ∗ in the dual of the auxiliary space we can associate an ordinary (k + 1)-formω

t∗on V , called the projection of ω along t∗, defined by

ωt∗ = 〈t∗, ω〉 . (1.29)

Obviously, ωt∗

depends linearly on t∗, so if we choose a basis e1, . . . , en of T , with dual

basis e1, . . . , en of T ∗, and define the ordinary (k + 1)-forms

ωa = ωea (1 6 a 6 n) , (1.30)

we obtainω = ωa ⊗ ea . (1.31)

Then it becomes clear thatker ω =

⋂t∗∈ T ∗

ker ωt∗ , (1.32)

and similarly

ker ω =n⋂

a=1

ker ωa , (1.33)

3The symbol 〈. , .〉 will be used to denote the natural bilinear pairing between a linear space and its dual.


and if we take the annihilator, that

supp ω =∑

t∗∈ T ∗

supp ωt∗ , (1.34)

and similarly

supp ω =n∑

a=1

supp ωa . (1.35)

The following characterization of the support is often used as a definition:

Lemma 1.1 The support supp ω of ω is the smallest subspace of V ∗ which satisfies

ω ∈(∧k+1

supp ω)⊗ T .

Proof. Using a basis e1, . . . , en of T as before, together with a basis e1, . . . , erof V composed of a basis e1, . . . , es of ker ω followed by a basis es+1, . . . , er of asubspace of V complementary to ker ω, and considering the corresponding dual basise1, . . . , er of V ∗, we see that es+1, . . . , er is a basis of supp ω = (ker ω)⊥, so expandingω in terms of the basis ep1∧ . . . ∧epk+1 ⊗ ea | 1 6 p1 < . . . < pk+1 6 r , 1 6 a 6 n of∧k+1

V ∗⊗ T , we conclude that all the coefficients ωap1...pk+1

with p1 6 s must vanish, which

implies ω ∈(∧k+1

supp ω)⊗ T . On the other hand, if Sω is any subspace of V ∗ such that

ω ∈(∧k+1

Sω

)⊗ T , then for every v ∈S⊥

ω we have ivω = 0 and hence v ∈ ker ω, that is,

S⊥ω ⊂ ker ω, implying that supp ω = (ker ω)⊥ ⊂Sω. 2

For forms of arbitrary degree k + 1 (k > 1), the notion of orthogonal complement ofa subspace depends on an additional parameter. Extending the definition given in Ref. [3]from ordinary to vector-valued forms, suppose that L is a subspace of V and ` is an integersatisfying 1 6 ` 6 k; then the `-orthogonal complement of L (with respect to ω) isdefined to be the subspace Lω,` of V given by4

Lω,` = v ∈V | iviv1. . . iv`

ω = 0 for all v1, . . . , v` ∈L . (1.36)

Note that these orthogonal complements are related by the following sequence of inclusions:

Lω,1 ⊂ . . . ⊂ Lω,`−1 ⊂ Lω,` ⊂ Lω,`+1 ⊂ . . . ⊂ Lω,k . (1.37)

As in the case of 2-forms, the notion of orthogonal complement can be used to define variouskinds of special subspaces:

4We discard the trivial case ` = 0 since extrapolating the definition given here would lead to the conclusionthat Lω,0 is simply the kernel of ω, independently of the subspace L of V .


• L is `-isotropic (with respect to ω) if L ⊂ Lω,` ;

• L is `-coisotropic (with respect to ω) if L ⊃ Lω,` ;

• L is `-lagrangian (with respect to ω) if it is both `-isotropic and `-coisotropic, that is,if L = Lω,` .

Among the `-isotropic subspaces, we distinguish the maximal ones, which are defined asusual: L is called maximal `-isotropic (with respect to ω) if L is `-isotropic (with respectto ω) and such that for any subspace L′ of V which is also `-isotropic (with respect to ω)

L ⊂ L′ =⇒ L = L′ .

Observe that both `-lagrangian and maximal `-isotropic subspaces contain the kernel of ω.

The following theorem assures that the concepts of `-lagrangian and maximal `-isotropicsubspaces are equivalent and establishes relations between such subspaces for a vector-valuedform and for its projections.

Theorem 1.1 Let V and T be finite-dimensional vector spaces and let ω be a T -valued(k + 1)-form on V . For a subspace L of V and 1 6 ` 6 k, we have

1. L is maximal `-isotropic with respect to ω iff L is `-lagrangian with respect to ω, andfor all t∗ ∈ T ∗, L is maximal `-isotropic with respect to ω

t∗iff L is `-lagrangian with

respect to ωt∗;

2. L is `-isotropic with respect to ω iff, for all t∗ ∈ T ∗, L is `-isotropic with respect to ωt∗;

3. If, for all t∗ ∈ T ∗, L is maximal `-isotropic with respect to ωt∗, then L is maximal

`-isotropic with respect to ω.

It should be emphasized that the converse to the last statement is false: a subspace whichis maximal isotropic with respect to a vector-valued form will be isotropic with respect toall of its projections but may fail to be maximal isotropic with respect to some of them.

Proof.

1. If L is `-isotropic, then for any vector u ∈Lω,` \ L, the subspace L′ = L + 〈u〉 of V is`-isotropic as well, since for ui = vi + λiu ∈L′ with vi ∈L (i = 0, . . . , `), we have

iu0iu1

. . . iu`ω =

∑i=0

iv0iv1

. . . iv`ω +

∑i=0

(−1)i λi iuiv0. . . ivi

. . . iv`ω = 0 .


Therefore, if L is maximal `-isotropic, then L′ ⊂L and hence Lω,` ⊂L , which impliesthat L is `-lagrangian. Conversely, if L is `-lagrangian and L′ is `-isotropic with L ⊂L′,then

L′ ⊂ (L′)ω,` ⊂ Lω,` = L ,

that is, L is maximal `-isotropic. The same argument can be applied to each of theprojected forms ω

t∗(t∗∈ T ∗), instead of ω.

2. For vi ∈L (i = 0, . . . , `), we have

iu0. . . iu`

ω = 0 ⇐⇒ iu0. . . iu`

ωt∗ = 0 for all t∗ ∈ T ∗ .

3. Suppose L is maximal `-isotropic with respect to each of the projected forms ωt∗

(t∗∈ T ∗). According to the previous item, L is `-isotropic with respect to ω. Moreover,if L′ is a subspace of V which is `-isotropic with respect to ω and such that L ⊂L′, thenaccording to the previous item, L′ is `-isotropic with respect to each of the projectedforms ω

t∗(t∗∈ T ∗) and hence L′ = L, proving that L is in fact maximal `-isotropic with

respect to ω.

2

In what follows, we are mainly interested in the case ` = 1 , so in order to simplify thenotation, we shall omit the prefix “1” whenever convenient.

Another way to approach the concept of (maximal) isotropic subspaces for vector-valuedforms which will turn out to be particularly fruitful is based on the idea of contracting ωwith vectors. Recall that contraction with ω is defined to be the linear map

ω[ : V −→(∧k

V ∗)⊗ T

v 7−→ ivω. (1.38)

Then it is clear that a subspace L of V will be isotropic if and only if

ω[(L)⊂(∧k

L⊥)⊗ T , (1.39)

which can be trivially rewritten in the form

ω[(L)⊂ ω[

(V)∩(∧k

L⊥)⊗ T . (1.40)

The merit of this – at first sight pointless – reformulation lies in its close analogy with thecriterion that a subspace L of V containing ker ω[ = ker ω will be maximal isotropic if andonly if

ω[(L)

= ω[(V)∩(∧k

L⊥)⊗ T . (1.41)


In fact, for any subspace L of V , we have the identity

ω[(Lω,1

)= ω[

(V)∩(∧k

L⊥)⊗ T , (1.42)

since both sides are equal to the subspace of(∧k

V ∗)⊗ T consisting of elements of the formivω where v ∈V and iuivω = 0 for all u ∈L.

Taking into account these considerations, we are led to introduce a concept that turnsout to provide the key to the theory of polysymplectic forms: the existence of a special typeof maximal isotropic subspace, which we shall call a polylagrangian subspace.

Definition 1.1 Let V and T be finite-dimensional vector spaces and let ω be a T -valued(k + 1)-form on V . We say that ω is a polylagrangian form of rank N if V admits asubspace L of codimension N which is polylagrangian, i.e., such that

ω[(L)

=(∧k

L⊥)⊗ T . (1.43)

When k = 1, we call ω a polysymplectic form.

The next theorem shows that a polylagrangian subspace, when it exists, really is a specialtype of maximal isotropic subspace; in particular, it contains the kernel of ω.

Theorem 1.2 Let V and T be finite-dimensional vector spaces, with n ≡ dim T , and letω be a T -valued polylagrangian (k+1)-form on V of rank N , with polylagrangian subspace L.Then N > k (except if ω ≡ 0) and L contains the kernel of ω as well as the kernel of eachof the projected forms ω

t∗(t∗∈ T ∗ \ 0):

ker ω ⊂ L , ker ωt∗ ⊂ L for all t∗∈ T ∗ \ 0 . (1.44)

The dimension of L is given by

dim L = dim ker ω + n

(N

k

). (1.45)

Proof. First we observe that if N < k, we have∧k

L⊥ = 0 and thus both sides of theequation (1.43) vanish; therefore, L is contained in ker ω and so ker ω has codimension < kin V , implying ω ≡ 0 , since the (k + 1)-form on the quotient space V/ ker ω induced by ωvanishes identically. (More generally, this argument shows that a nonvanishing vector-valued(k + 1)-form does not permit isotropic subspaces of codimension < k.) Thus supposing thatdim L⊥ = N > k, we can for any vector v ∈ V \L find a linearly independent set of 1-formsv∗1, . . . , v

∗k ∈L⊥ such that 〈v∗1, v〉 = 1 and 〈v∗i , v〉 = 0 for i > 1. Given t∗∈ T ∗, take t ∈ T


such that 〈 t∗, t 〉 = 1. According to the definition of a polylagrangian subspace, there is avector u ∈L such that

iuω = v∗1 ∧ . . . ∧ v∗k ⊗ t ⇒ iviuωt∗ = v∗2 ∧ . . . ∧ v∗k 6= 0

and so v /∈ ker ωt∗

. Hence it follows that ker ω ⊂ ker ωt∗⊂L, and we obtain

dim L − dim ker ω = dim ω[(L)

= dim((∧k

L⊥)⊗ T)

= n

(N

k

).

2

The dimension formula stated in the previous theorem provides a simple criterion to decidewhether a given isotropic subspace is polylagrangian.

Theorem 1.3 Let V and T be finite-dimensional vector spaces, with n ≡ dim T , and letω be a T -valued (k + 1)-form on V . If L is an isotropic subspace of V of codimension Ncontaining ker ω and such that

dim L = dim ker ω + n

(N

k

), (1.46)

then L is polylagrangian and ω is a polylagrangian form.

Proof. Since L is an isotropic subspace of V containing ker ω, the restriction of the linear

map ω[ (see equation (1.38)) to L takes L to(∧k

L⊥) ⊗ T and induces an injective linear

map from the quotient space L/ ker ω to(∧k

L⊥)⊗ T which, due to the condition (1.46),is a linear isomorphism, so the equality (1.43) follows. 2

In the case of truly vector-valued forms, we can say much more.

Theorem 1.4 Let V and T be finite-dimensional vector spaces, with n ≡ dim T > 2, andlet ω be a T -valued (k+1)-form on V which is polylagrangian of rank N , with polylagrangiansubspace L. Then L is given by

L =∑

t∗∈ T ∗\0

ker ωt∗ (1.47)

and, in particular, is unique. In terms of a basis e1, . . . , en of T with dual basise1, . . . , en of T ∗,

L = ker ω ⊕K1 ⊕ . . .⊕Kn , (1.48)

and for any 1 6 a 6 n,L = ker ωa ⊕Ka , (1.49)


where for any 1 6 a 6 n, Ka is any subspace of V satisfying

n⋂b=1b6=a

ker ωb = ker ω ⊕Ka . (1.50)

The dimensions of these various subspaces are given by

dim ker ωa = dim ker ω + (n− 1)

(N

k

), dim Ka =

(N

k

). (1.51)

Proof. Fix a basis e1, . . . , en of T with dual basis e1, . . . , en of T ∗ and choosesubspaces Ka of V (a = 1, . . . , n) as indicated above. Then the subspaces ker ω andK1, . . . , Kn of V have trivial intersection, so their sum is direct and defines a subspace of Vwhich we shall, for the moment, denote by L′. According to the previous theorem, L′ ⊂L.To show that L′ = L, it is therefore sufficient to prove that ω[

(L)⊂ ω[

(L′), since both L

and L′ contain ker ω. Using the definition of a polylagrangian subspace, we conclude thatwe must establish the inclusion (∧k

L⊥)⊗ T ⊂ ω[(L′) .

But the equality (1.43) guarantees that for any α ∈∧k

L⊥ and for any 1 6 a 6 n, there is avector va ∈L such that

ivaω = α⊗ ea .

Sinceiva

ωb = α 〈eb, ea〉 = δba α ,

we see that

va ∈n⋂

b=1b6=a

ker ωb .

Decomposing va according to equation (1.50), we find a vector ua ∈Ka such that

iuaω = α⊗ ea ,

so α ⊗ ea ∈ ω[(Ka) ⊂ ω[(L′). Finally, we observe that, due to the definition (1.50) of thespaces Ka, together with the relation (1.33), we have

ker ωa ∩Ka = 0

so that equation (1.49) follows from equation (1.48), whereas equation (1.51) is a directconsequence of the formulas (1.45) or (1.46), (1.48) and (1.49). 2

But even for scalar forms, i.e., when dim T = 1, uniqueness of the polylagrangian subspaceis assured, except for 2-forms (k = 1).

1.2 Partially horizontal forms on linear spaces 13

Proposition 1.1 Let V be a finite-dimensional vector space and ω a polylagrangian (k+1)-form of rank N on V , with polylagrangian subspace L. If k > 1, L is unique.

A proof can be found in Ref. [19] (the additional hypothesis made there that ω should benon-degenerate can easily be removed if we replace V and L by the respective quotient spacesV/ ker ω and L/ ker ω).

In the case of symplectic forms (dim T = 1, k = 1, ker ω = 0), where the concept of apolylagrangian subspace reduces to the familiar concept of a lagrangian subspace, it is wellknown that such a uniqueness statement does not hold. The fundamental importance ofpolylagrangian subspaces in the process of generalizing lagrangian subspaces from the sym-plectic case to vector-valued forms stems from the fact that they are essential for obtaininga Darboux theorem.

Ordinary non-degenerate forms of degree > 2 (dim T = 1, k > 1, ker ω = 0) havebeen studied in the literature [3, 19] under the label “multisymplectic forms”, but withoutemphasizing the central role played by the concept of a polylagrangian subspace, whichappears only implicitly through the dimension criterion formulated in Theorem 1.3, employedas a definition. Anyway, the restriction imposed by the existence of such a subspace is anecessary condition to obtain a “Darboux basis” for the form ω.

1.2 Partially horizontal forms on linear spaces

In this section we present concepts analogous to those introduced in the previous section, butfor ordinary forms subject to a condition of (partial) horizontality relative to a given verticalsubspace, which requires some modifications. Once again, we introduce a special type ofmaximal isotropic subspace which will be called a multilagrangian subspace and which playsa fundamental role in the theory of multisymplectic forms.

Let W be a finite-dimensional vector space and V be a fixed subspace of W , the verticalspace. Following the notation already employed at the beginning of this chapter, we denotethe base space W/V by T , setting dim T = n, and the canonical projection of W onto Tby π, thus obtaining the exact sequence (1.3). Moreover, suppose that ω is a (k + 1)-formon W which is (k + 1− r)-horizontal, where 1 6 r 6 k and k + 1− r 6 n : 5

ω ∈∧k+1

r W ∗ . (1.52)

5The trivial cases r = 0 and r = k + 1 will be excluded since they are already covered by the formalismdeveloped in the previous section, with dim T = 1, whereas the additional condition k + 1− r 6 n (whichneed not be stated separately if k 6 n) can be imposed without loss of generality since otherwise, ω willhave to vanish identically.


As we are dealing with a special case of the situation studied before, the definitions givenat the beginning of the previous section can be applied without modification: this includesthe concepts of kernel and of support, the notion of a non-degenerate form, the idea ofrank and the definition of the `-orthogonal complement of a subspace. The first importantdifference occurs in the study of isotropic subspaces, since in the case of partially horizontalforms, we shall consider only isotropic subspaces L of the total space W contained in thevertical subspace V , which leads to a concept of maximality different from the previousone: a subspace L of V will be called maximal isotropic in V (with respect to ω) if L isisotropic (with respect to ω) and such that for any subspace L′ of V which is also isotropic(with respect to ω)

L ⊂ L′ =⇒ L = L′ .

Using the restriction of the linear map

ω[W : W −→

∧kW ∗

w 7−→ iwω

defined by equation (1.38) to the vertical subspace V , which is a linear map

ω[V : V −→

∧ kr−1W

∗

v 7−→ ivω, (1.53)

and defining, for any subspace L of V ,∧ kr−1 L⊥ =

∧kL⊥ ∩

∧ kr−1W

∗ , (1.54)

we conclude that a subspace L of V will be isotropic if and only if

ω[V

(L)⊂∧ k

r−1 L⊥ , (1.55)

which can be trivially rewritten in the form

ω[V

(L)⊂ ω[

V

(V)∩∧ k

r−1 L⊥ . (1.56)

Similarly, we conclude that a subspace L of V containing ker ω[V = V ∩ ker ω will be

maximal isotropic in V if and only if

ω[V

(L)

= ω[V

(V)∩∧ k

r−1 L⊥ . (1.57)

In fact, for any subspace L of V , we have the identity

ω[V

(Lω,1 ∩V

)= ω[

V

(V)∩∧ k

r−1 L⊥ , (1.58)

1.2 Partially horizontal forms on linear spaces 15

since both sides are equal to the subspace of∧ k

r−1W∗ consisting of elements of the form

ivω where v ∈V and iuivω = 0 for all u ∈L.

As in the case of polylagrangian forms, these considerations lead us to introduce a conceptthat turns out to provide the key to the theory of multisymplectic forms: the existence of aspecial type of maximal isotropic subspace, which we shall call a multilagrangian subspace.

Definition 1.2 Let W be a finite-dimensional vector space and V be a fixed subspaceof W , with n = dim(W/V ), and let ω be a (k +1− r)-horizontal (k +1)-form on W , where1 6 r 6 k and k + 1− r 6 n. We say that ω is a multilagrangian form of rank N anddegree of horizontality k + 1 − r if V admits a subspace L of codimension N which ismultilagrangian, i.e., such that

ω[V

(L)

=∧ k

r−1 L⊥ . (1.59)

When k = n, r = 2 and ω is non-degenerate, we call ω a multisymplectic form.

The following theorem shows that a multilagrangian subspace, when it exists, really is aspecial type of maximal isotropic subspace of V ; in particular, it contains the kernel of ω.

Theorem 1.5 Let W be a finite-dimensional vector space and V be a fixed subspace of W ,with n = dim(W/V ), and let ω be a multilagrangian (k+1)-form on W of rank N and degreeof horizontality k + 1 − r, where 1 6 r 6 k and k + 1 − r 6 n, and with multilagrangiansubspace L. Then N + n > k (except if ω ≡ 0) and L contains the kernel of ω:

ker ω ⊂ L . (1.60)

The dimension of L is given by

dim L = dim ker ω +r−1∑s=0

(N

s

)(n

k − s

), (1.61)

where it is understood that, by definition,(

pq

)= 0 if q > p .

Proof. First we observe that if N + n < k, we have∧k

L⊥ = 0 and thus both sides ofthe equation (1.59) vanish; therefore, L is contained in ker ω and so ker ω has codimension< k in W , implying ω ≡ 0 , since the (k+1)-form on the quotient space W/ ker ω induced byω vanishes identically. (More generally, this argument shows that a nonvanishing (k+1−r)-horizontal (k + 1)-form does not permit isotropic subspaces of codimension < k − n.) Thussupposing that dim L⊥ = N + n > k and using that dim V ⊥ = n and V ⊥ ⊂L⊥, weconclude that we can, for any vector w ∈ W \ L, find a linearly independent set of 1-forms


w∗1, . . . , w

∗k ∈L⊥ such that w∗

r , . . . , w∗k ∈V ⊥ and 〈w∗

1, w〉 = 1 and 〈w∗i , w〉 = 0 for i > 1.

According to the definition of a multilagrangian subspace, there is a vector u ∈L such that

iuω = w∗1 ∧ . . . ∧ w∗

k ⇒ iwiuω = w∗2 ∧ . . . ∧ w∗

k 6= 0

and so w /∈ ker ω. Hence it follows that ker ω ⊂L, and we obtain

dim L − dim ker ω = dim ω[V

(L)

= dim∧ k

r−1 L⊥ .

To calculate this dimension we proceed as at the beginning of this chapter, introducing abasis eL

1 , . . . , eLl , eL′

1 , . . . , eL′N , eT

1 , . . . , eTn of W such that the first l vectors form a basis

of L, the following N vectors form a basis of a subspace L′ complementary to L in V andthe last n vectors form a basis of a subspace H complementary to V in W , isomorphicto T . Then in terms of the dual basis e1

L, . . . , elL, e1

L′ , . . . , eNL′ , e

1T , . . . , en

T of W ∗, weconclude that ei1

L′ ∧ . . . ∧ eisL′ ∧ eµ1

T ∧ . . . ∧ eµk−s

T | 0 6 s 6 r − 1 , 1 6 i1 < . . . < is 6 N ,

1 6 µ1 < . . . < µk−s 6 n is a basis of∧ k

r−1 L⊥ , which proves the formula (1.61). 2

The dimension formula stated in the previous theorem provides a simple criterion to decidewhether a given isotropic subspace is multilagrangian.

Theorem 1.6 Let W be a finite-dimensional vector space and V be a fixed subspace of W ,with n = dim(W/V ), and let ω be a (k + 1 − r)-horizontal (k + 1)-form on W , where1 6 r 6 k and k + 1 − r 6 n. If L is an isotropic subspace of V of codimension Ncontaining ker ω and such that

dim L = dim ker ω +r−1∑s=0

(N

s

)(n

k − s

), (1.62)

where it is understood that, by definition,(

pq

)= 0 if q > p, then L is a multilagrangian

subspace and ω is a multilagrangian form.

Proof. Since L is an isotropic subspace of V containing ker ω, the restriction of the

linear map ω[V (see equation (1.53)) to L takes L to

∧ kr−1 L⊥ and induces an injective linear

map from the quotient space L/ ker ω into∧ k

r−1 L⊥ which, due to the condition (1.62), is

a linear isomorphism, so the equality (1.59) follows. 2

A particularly simple case occurs when r = 1, since every k-horizontal multilagrangian(k + 1)-form has rank 0 and multilagrangian subspace V . (In fact, the condition that ωshoud be k-horizontal is equivalent to the condition that V should be isotropic, and in thiscase, V does satisfy the remaining criteria of Theorem 1.6.)

1.3 The symbol 17

1.3 The symbol

Our main goal in this section is to exhibit a simple and general relation between multi-lagrangian and polylagrangian forms: the symbol of a multilagrangian (multisymplectic)form is a polylagrangian (polysymplectic) form taking values in a space of forms of appro-priate degree on the base space.

To explain this construction, we adopt the same notation as in the previous section,that is, W is a finite-dimensional vector space and V is a fixed subspace of W , the verticalspace, whereas the quotient space W/V , the base space, will be denoted by T , its dimensionby n = dim T and the canonical projection from W to T by π, giving rise to the exactsequence (1.3). We shall also suppose that ω is a (k + 1− r)-horizontal (k + 1)-form on W ,where 1 6 r 6 k and k + 1− r 6 n :

ω ∈∧k+1

r W ∗ . (1.63)

Then the symbol ω of ω, defined as at the beginning of this chapter, will be a∧k+1−r

T ∗ -valued r-form on V :

ω ∈∧r

V ∗ ⊗∧k+1−r

T ∗ . (1.64)

Using the canonical isomorphism∧k+1−r

T ∗ ∼=∧k+1−r

0 W ∗ as an identification, we have

ω(v1, . . . , vr) = ivr. . . iv1

ω for v1, . . . , vr ∈V . (1.65)

More explicitly, if we use a splitting s of π, we have

ω(v1, . . . , vr)(t1, . . . , tk+1−r

)= ω(v1, . . . , vr, s(t1), . . . , s(tk+1−r))

for v1, . . . , vr ∈V , t1, . . . , tk+1−r ∈T ,(1.66)

as has already been mentioned at the beginning of this chapter.

A straightforward analysis of the formula (1.66) shows that a subspace L of the verticalspace V which is isotropic with respect to ω will also be isotropic with respect to ω (theconverse statement is not true). Moreover, we have

Theorem 1.7 Let W be a finite-dimensional vector space and V be a fixed subspace of W ,with n = dim(W/V ), let ω be a (k + 1− r)-horizontal (k + 1)-form on W , where 1 6 r 6 k

and k + 1 − r 6 n, and let ω be the symbol of ω, which is a∧k+1−r

T ∗-valued r-formon V . Suppose that ω is multilagrangian, with multilagrangian subspace L. Then ω is poly-lagrangian, with polylagrangian subspace L. If ω is multisymplectic then ω is polysymplectic.


Proof. Fixing an arbitrary horizontal subspace H of W and using the direct decomposi-tions (1.17) and (1.18), with H⊥ ∼= V ∗, we note that in order to show that L is polylagrangianwith respect to ω, we must establish the equality

ω[(L) ∼= ∧r−1

(L⊥ ∩H⊥) ⊗∧k+1−r

T ∗ .

To do so, we use the canonical isomorphism∧k+1−r

T ∗ ∼=∧k+1−r

0 W ∗ =∧k+1−r

V ⊥ and

the inclusion V ⊥ ⊂L⊥, together with the fact that the space∧r−1

(L⊥ ∩H⊥) ⊗∧k+1−r

V ⊥

is generated by elements which can be written in the form

α =(w∗

1 ∧ . . . ∧ w∗r−1

)⊗(w∗

r ∧ . . . ∧ w∗k

)where α = w∗

1 ∧ . . . ∧ w∗k

with w∗1, . . . , w

∗r−1 ∈ L⊥ ∩ H⊥ and w∗

r , . . . , w∗k ∈ V ⊥ ⊂L⊥. Since L is multilagrangian with

respect to ω, there is a vector u ∈L such that α = iuω and so α = iuω, showing that L ispolylagrangian with respect to ω. 2

We have shown that the multilagrangian subspace associated with a multilagrangian form ωis the same as the polylagrangian subspace associated with its symbol ω, but it is worthwhileto note that the kernels of the two forms need not coincide: The only relation that can beestablished in general is the inclusion

ker ω ⊂ ker ω . (1.67)

In particular, it may (and often does) happen that ω is non-degenerate while ω is degen-erate. In what follows, we shall frequently consider the projections ωt1,...,tk+1−r

of ω along

decomposable tensors t1 ∧ . . . ∧ tk+1−r ∈∧k+1−r

T : then using a splitting s of π, we will havefor the respective kernels

ker ωt1,...,tk+1−r= v ∈V | ω

(v, v1, . . . , vr−1, s(t1), . . . , s(tk+1−r)

)= 0

para v1, . . . , vr−1 ∈V , t1, . . . , tk+1−r ∈T , (1.68)

which implies that, as a consequence of the results obtained in the first section, ker ω is theintersection of all these subspaces and, for 0 < k + 1− r < n, L is their sum.

1.4 Canonical Examples

In this section we shall introduce the canonical examples of a polylagrangian and of a multi-lagrangian vector space. As we shall see later, every non-degenerate polylagrangian or multi-lagrangian form is equivalent to the canonical one presented here, that is, there is a linearisomorphism between the underlying vector spaces relating them by pull-back. Throughout

1.4 Canonical Examples 19

the entire discussion, we shall fix the integer k that characterizes the degree of the form (whichis k + 1, with k > 1) and, in the multilagrangian case, the integer r that characterizes thehorizontal degree of the form (which is k + 1 − r, with 1 6 r 6 k and k + 1 − r 6 n, asbefore), in order to avoid overloading the notation with prefixes such as “k-” or “(k, r)-”.

1.4.1 The canonical polylagrangian form

Let E and T be vector spaces of dimension N and n, respectively. Set

V0 = E ⊕((∧k

E∗)⊗ T)

. (1.69)

Definition 1.3 The canonical polylagrangian form of rank N is the non-degenerateT -valued (k + 1)-form ω0 on V0 defined by

ω0

((u0, α0 ⊗ t0), . . . , (uk, αk ⊗ tk)

)=

k∑i=0

(−1)i αi(u0, . . . , ui, . . . , uk) ti . (1.70)

If k = 1 we shall call ω0 the canonical polysymplectic form.

For scalar forms (T = R) this construction can be found, e.g., in Refs [19] or [3].

To prove the statements contained in the above definition, note that it is a straightforwardexercise to show that ω0 is non-degenerate and that, considering E and

L =(∧k

E∗)⊗ T (1.71)

as subspaces of V0, we have the direct decomposition V0 = E ⊕ L where

L is polylagrangian and E is k-isotropic , (1.72)

since L is obviously isotropic and has the right dimension, as required by Theorem 1.3:

dim L = n

(N

k

)where codimV0

L = N . (1.73)

In terms of bases, let ea | 1 6 a 6 n be any basis of T with dual basis ea | 1 6 a 6 n of T ∗ and let ei | 1 6 i 6 N be any basis of E with dual basis ei | 1 6 i 6 N of E∗.For 1 6 a 6 n and 1 6 i1 < . . . < ik 6 N , define

ei1... ika = ei1∧ . . . ∧ eik ⊗ ea , ea

i1... ik= ei1

∧ . . . ∧ eik⊗ ea .


This provides a basis ei, ei1... ika | 1 6 a 6 n , 1 6 i 6 N , 1 6 i1 < . . . < ik 6 N of V0

with dual basis ei, eai1... ik

| 1 6 a 6 n , 1 6 i 6 N , 1 6 i1 < . . . < ik 6 N of V ∗0 , both of

which we shall refer to as a Darboux basis, such that

ω0 = 1k!

(ea

i1... ik∧ ei1 ∧ . . . ∧ eik

)⊗ ea . (1.74)

In Section 1.5 we shall see that any polylagrangian vector space splits into the direct sum ofa (its) polylagrangian subspace and a second component which is k-isotropic, as above. Thisproperty is precisely what is needed to show that every polylagrangian form is equivalent tothe canonical one – a statement that can be viewed as a basis independent formulation of(the algebraic version of) the Darboux Theorem for polylagrangian vector spaces.

1.4.2 The canonical multilagrangian form

Let F be a vector space of dimension N + n and E be a fixed N -dimensional subspace of F .Denoting the n-dimensional quotient space F/E by T and the canonical projection of Fonto T by ρ, we obtain the following exact sequence of vector spaces:

0 −→ E −→ Fρ−→ T −→ 0 . (1.75)

DefineW0 = F ⊕

∧ kr−1F

∗ , V0 = E ⊕∧ k

r−1F∗ , π0 = ρ pr1 , (1.76)

where pr1 : W0 → F is the canonical projection, which leads us to the following exactsequence of vector spaces:

0 −→ V0 −→ W0

π0−→ T −→ 0 . (1.77)

Definition 1.4 The canonical multilagrangian form of rank N and horizontal degreek + 1− r is the (k + 1− r)-horizontal (k + 1)-form ω0 on W0 defined by

ω0

((u0, α0), . . . , (uk, αk)

)=

k∑i=0

(−1)i αi(u0, . . . , ui, . . . , uk) . (1.78)

If k = n and r = 2 we shall call ω0 the canonical multisymplectic form.

Note that when r = 1, the form ω0 is degenerate, with

ker ω0 = E if r = 1 , (1.79)

whereas for r > 1, it is non-degenerate. In fact, suppose that (u, α) ∈ W0 \ 0 . If α 6= 0,there exist vectors u1, . . . , uk ∈F such that

ω0

((u, α), (u1, 0), . . . , (uk, 0)

)= α(u1, . . . , uk) 6= 0

1.4 Canonical Examples 21

and hence (u, α) /∈ ker ω0 . If r = 1 and u /∈E or if r > 1 and u 6= 0, there exists a

(k + 1 − r)-horizontal k-form β ∈∧ k

r−1F∗ such that iuβ 6= 0 ; hence we can find vectors

u2, . . . , uk ∈F such that

ω0((u, 0), (0, β), (u2, 0), . . . , (uk, 0)) = − β(u, u2, . . . , uk) 6= 0 ,

that is, (u, 0) /∈ ker ω0 .

In what follows, we shall assume that N > 0, since when E = 0, we are back to the

polylagrangian case on T ⊕∧k

T ∗, with T = R, which has already been studied in Refs [19]and [3]. For the same reason, we shall also assume that r > 1, since for r = 1 we have∧ k

0F∗ ∼=

∧kT ∗, so that after passing to the quotient by the kernel of ω0, we are once again

back to the polylagrangian case on T ⊕∧k

T ∗, with T = R.

To prove the statements contained in the above definition (under the additional hypo-theses that N > 0 and r > 1), note that it is a straightforward exercise to show that ω0 is(k + 1− r)-horizontal and that, considering F and

L =∧ k

r−1F∗ (1.80)

as subspaces of W0, we have the direct decompositions W0 = F ⊕L and V0 = E⊕L where

L is multilagrangian , F is k-isotropic , E is (r − 1)-isotropic , (1.81)

since L is obviously isotropic and has the right dimension, as required by Theorem 1.6:

dim L =r−1∑s=0

(N

s

)(n

k − s

)where codimV0

L = N . (1.82)

In terms of bases, let ei, eµ | 1 6 i 6 N , 1 6 µ 6 n be a basis of F with dual basis ei, eµ | 1 6 i 6 N , 1 6 µ 6 n of F ∗ such that ei | 1 6 i 6 N is a basis of E and eµ | 1 6 µ 6 n is a basis of a subspace H of F complementary to E, isomorphic to T .For 0 6 s 6 r, 1 6 i1 < . . . < is 6 N and 1 6 µ1 < . . . < µk−s 6 N , define

ei1... is; µ1... µk−s= ei1

∧ . . . ∧ eis ∧ eµ1∧ . . . ∧ eµk−s

ei1... is; µ1... µk−s = ei1 ∧ . . . ∧ eis ∧ eµ1 ∧ . . . ∧ eµk−s.

This provides a basis

ei, eµ, ei1... is; µ1... µk−s | 0 6 s 6 r − 1 ,

1 6 i 6 N , 1 6 i1 < . . . < is 6 N1 6 µ 6 n , 1 6 µ1 < . . . < µk−s 6 n


of W0 with dual basis

ei, eµ, ei1... is; µ1... µk−s| 0 6 s 6 r − 1 ,

1 6 i 6 N , 1 6 i1 < . . . < is 6 N1 6 µ 6 n , 1 6 µ1 < . . . < µk−s 6 n

of W ∗0 , both of which we shall refer to as a Darboux basis, such that

ω0 =r−1∑s=0

1s!

1(k−s)!

ei1... is; µ1... µk−s∧ ei1 ∧ . . . ∧ eis ∧ eµ1 ∧ . . . ∧ eµk−s . (1.83)

and for the symbol

ω0 = 1(r−1)!

1(k+1−r)!

(ei1... ir−1; µ1... µk+1−r

∧ ei1 ∧ . . . ∧ eir−1)⊗(eµ1 ∧ . . . ∧ eµk+1−r

). (1.84)

In Section 1.6 we shall see that every multilagrangian vector space splits into the direct sumof a (its) multilagrangian subspace and a second component which is k-isotropic and containsan (r − 1)-isotropic subspace such that the quotient of these two components is isomorphicto the base space T , as above. This property is precisely what is needed to show that everymultilagrangian form is equivalent to the canonical one – a statement that can be viewedas a basis independent formulation of (the algebraic version of) the Darboux Theorem formultilagrangian vector spaces.

1.5 Polylagrangian vector spaces

In this section we will show that every polylagrangian vector space admits a Darboux basis,or canonical basis, in terms of which the polylagrangian form is given by the standardexpression of equation (1.74).

Definition 1.5 Let V and T be finite-dimensional vector spaces, with n ≡ dim T , and let ωbe a T -valued polylagrangian (k + 1)-form on V of rank N , with polylagrangian subspace L.A (polylagrangian) Darboux basis or (polylagrangian) canonical basis for ω isa basis ei, e

i1... ika | 1 6 a 6 n , 1 6 i 6 N , 1 6 i1 < . . . < ik 6 N of any sub-

space of V complementary to ker ω, with dual basis ei, eai1... ik

| 1 6 a 6 n , 1 6 i 6 N ,1 6 i1 < . . . < ik 6 N of the subspace supp ω = (ker ω)⊥ of V ∗, together with a basis ea | 1 6 a 6 n of T , with dual basis ea | 1 6 a 6 n of T ∗, such that

ω = 1k!

(ea

i1... ik∧ ei1 ∧ . . . ∧ eik

)⊗ ea . (1.85)

If k = 1 we shall call such a basis a (polysymplectic) Darboux basis or (poly-symplectic) canonical basis.

1.5 Polylagrangian vector spaces 23

The existence of such a basis for any polylagrangian form follows from the existence of adirect decomposition V = L⊕ E of V analogous to the one for V0 in Section 1.4.1.

Theorem 1.8 Let V and T be finite-dimensional vector spaces, with n ≡ dim T , and letω be a T -valued polylagrangian (k+1)-form on V of rank N , with polylagrangian subspace L.Then there exists a k-isotropic subspace E of V complementary to L, i.e., such that

V = E ⊕ L . (1.86)

Proof. Let E0 be a k-isotropic subspace of V of dimension N ′ such that E0 ∩ L = 0.(For instance, as long as N ′ 6 k, E0 can be any subspace of V such that E0 ∩ L = 0.)If N ′ = N , we are done. Otherwise, choose a basis e1, . . . , eN of a subspace of V com-plementary to L such that the first N ′ vectors constitute a basis of E0, and denote thecorresponding dual basis of L⊥ by e1, . . . , eN. We shall prove that there exists a vectoru ∈V \ (E0 ⊕ L) such that the subspace E1 of V spanned by u and E0 is k-isotropic andsatisfies E1 ∩ L = 0; then since dim E1 = N ′ + 1, the statement of the theorem followsby induction. To this end, consider an arbitrary basis ea | 1 6 a 6 n of T with dualbasis ea | 1 6 a 6 n of T ∗ and, choosing any subspace L′ of L complementary toker ω, use the fact that L is polylagrangian to conclude that there exists a unique basis ei1... ik

a | 1 6 a 6 n , 1 6 i1 < . . . < ik 6 N of L′ such that

ω[(ei1... ika ) = ei1 ∧ . . . ∧ eik ⊗ ea .

Thus, for 1 6 i1 < . . . < ik 6 N and 1 6 j1 < . . . < jk 6 N , we have

ωb(ei1... ika , ej1

, . . . , ejk) = δb

a δi1j1

. . . δikjk

.

Therefore, the vector

u = eN ′+1 − 1k!

ωa(eN ′+1, ei1, . . . , eik

) ei1... ika

does not belong to the subspace E0 ⊕ L and, for 1 6 j1 < . . . < jk 6 N , satisfies

ωb(u, ej1, . . . , ejk

)

= ωb(eN ′+1, ej1, . . . , ejk

) − 1k!

ωa(eN ′+1, ei1, . . . , eik

) ωb(ei1... ika , ej1

, . . . , ejk)

= 0 ,

which implies that since the subspace E0 spanned by e1, . . . , eN ′ is k-isotropic, the subspaceE1 spanned by e1, . . . , eN ′ and u is so as well. 2

Now it is easy to derive the algebraic Darboux theorem: let ea | 1 6 a 6 n be an arbitrarybasis of T , with dual basis ea | 1 6 a 6 n of T ∗, and let ei | 1 6 i 6 N be an arbitrary


basis of a k-isotropic subspace E complementary to L in V , with dual basis ei | 1 6 i 6 N of L⊥ ∼= E∗. Choosing an arbitrary subspace L′ of L complementary to ker ω and taking intoaccount the identity (1.43), we define a basis ei1... ik

a | 1 6 a 6 n , 1 6 i1 < . . . < ik 6 N of L′ by

ω[(ei1... ika ) = ei1 ∧ . . . ∧ eik ⊗ ea .

It is easy to see that the union of this basis with that of E gives a canonical basis of V (ormore precisely, of E⊕L′, which is a subspace of V complementary to ker ω). Thus we haveproved

Theorem 1.9 (Darboux theorem for polylagrangian vector spaces)

Every polylagrangian vector space admits a canonical basis.

Clearly, the inductive construction of a k-isotropic subspace E complementary to the poly-lagrangian subspace L, as presented in the proof of Theorem 1.8, provides an iterative andexplicit method to construct polylagrangian bases in a way similar to the well known Gram-Schmidt orthogonalization process.

1.6 Multilagrangian vector spaces

In this section we will show that every multilagrangian vector space admits a Darboux basis,or canonical basis, in termos of which the multilagrangian form is given by the standardexpression of equation (1.83).

Definition 1.6 Let W be a finite-dimensional vector space and V be a fixed subspace of W ,with n = dim(W/V ), and let ω be a multilagrangian (k+1)-form on W of rank N and degreeof horizontality k + 1 − r, where 1 6 r 6 k and k + 1 − r 6 n, and with multilagrangiansubspace L. A (multilagrangian) Darboux basis or (multilagrangian) canonicalbasis for ω is a basis

ei, eµ, ei1... is; µ1... µk−s | 0 6 s 6 r − 1 ,

1 6 i 6 N , 1 6 i1 < . . . < is 6 N1 6 µ 6 n , 1 6 µ1 < . . . < µk−s 6 n

of any subspace of W complementary to ker ω, with dual basis

ei, eµ, ei1... is; µ1... µk−s| 0 6 s 6 r − 1 ,

1 6 i 6 N , 1 6 i1 < . . . < is 6 N1 6 µ 6 n , 1 6 µ1 < . . . < µk−s 6 n

of the subspace supp ω = (ker ω)⊥ of W ∗, such that

ω =r−1∑s=0

1s!

1(k−s)!

ei1... is; µ1... µk−s∧ ei1 ∧ . . . ∧ eis ∧ eµ1 ∧ . . . ∧eµk−s . (1.87)

1.6 Multilagrangian vector spaces 25

In this same basis we have for the symbol

ω = 1(r−1)!

1(k+1−r)!

(ei1... ir−1; µ1... µk+1−r

∧ ei1 ∧ . . . ∧ eir−1)⊗(eµ1 ∧ . . . ∧ eµk+1−r

). (1.88)

If k = n and r = 2 we shall call such a basis a (multisymplectic) Darboux basis or(multisymplectic) canonical basis.

The existence of such a basis for any multilagrangian form follows from the existence ofdirect decompositions W = L⊕F of W and V = L⊕E of V analogous to the ones for W0

and V0 in Section 1.4.2.

Theorem 1.10 Let W be a finite-dimensional vector space and V be a fixed subspace of W ,with n = dim(W/V ), and let ω be a multilagrangian (k+1)-form on W of rank N and degreeof horizontality k + 1 − r, where 1 6 r 6 k and k + 1 − r 6 n, and with multilagrangiansubspace L. Then there exists a k-isotropic subspace F of W such that the intersectionE = V ∩F is an (r − 1)-isotropic subspace of V and

W = F ⊕ L , V = E ⊕ L . (1.89)

Proof. First we construct an (r − 1)-isotropic subspace E of V of dimension N which iscomplementary to L in V . If r = 1 there is nothing to prove since in this case the verticalsubspace V is isotropic and so we have L = V , E = 0 and N = 0 . If r > 1 we applyTheorem 1.8 to the symbol ω of ω to conclude that there is a subspace E of V of dimension Nwhich is complementary to L in V and is (r−1)-isotropic with respect to ω. Now taking intoaccount that the whole vertical subspace V is r-isotropic with respect to ω, it follows that Eis (r− 1)-isotropic with respect to ω as well. – Now let F0 be a subspace of W of dimensionN + n′ which is k-isotropic with respect to ω and such that F0 ∩ V = E. (For instance, ifn′ = 0, F0 = E). If n′ = n, we are done. Otherwise, choose a basis eE

1 , . . . , eEN , e1, . . . , en

of a subspace of W complementary to L such that the first N vectors constitute a basis of Eand the first N +n′ vectors constitute a basis of F0, and denote the corresponding dual basisof L⊥ by e1

E, . . . , eNE , e1, . . . , en. We shall prove that there exists a vector u ∈W \ (F0⊕L)

such that the subspace F1 spanned by u and F0 is k-isotropic and satisfies F1 ∩ V = E;then since dim F1 = N + n′ + 1, the statement of the theorem follows by induction. Tothis end, choose any subspace L′ of L complementary to ker ω and use the fact that L ismultilagrangian to conclude that there exists a unique basis

ei1... is; µ1... µk−s | 0 6 s 6 r − 1 ,1 6 i1 < . . . < is 6 N

1 6 µ1 < . . . < µk−s 6 n

of L′ such that

ω[V (ei1... is; µ1... µk−s) = ei1

E ∧ . . . ∧ eisE ∧ eµ1 ∧ . . . ∧ eµk+1−s .


Thus, for 0 6 s, t 6 r − 1, 1 6 i1 < . . . < is 6 N , 1 6 j1 < . . . < jt 6 N ,1 6 µ1 < . . . < µk−s 6 n, 1 6 ν1 < . . . < νk−t 6 n, we have

ω(ei1... is; µ1... µk−s , eEj1

, . . . , eEjt, eν1

, . . . , eνk−t) =

0 if s 6= t

δi1j1

. . . δisjs

δµ1ν1

. . . δµk−sνk−s if s = t

.

Therefore, the vector

u = en′+1 −r−1∑s=0

1s!

1(k−s)!

ω(en′+1, eEi1, . . . , eE

is , eµ1, . . . , eµk−s

) ei1... is; µ1... µk−s

does not belong to the subspace F0 ⊕ L and, for 0 6 t 6 r − 1, 1 6 j1 < . . . < jt 6 N and1 6 ν1 < . . . < νk−t 6 n, satisfies

ω(u, eEj1

, . . . , eEjt, eν1

, . . . , eνk−t)

= ω(en′+1, eEj1

, . . . , eEjt, eν1

, . . . , eνk−t)

−r−1∑s=0

1s!

1(k−s)!

ω(en′+1, eEi1, . . . , eE

is , eµ1, . . . , eµk−s

)

× ω(ei1... is; µ1... µk−s , eEj1

, . . . , eEjt, eν1

, . . . , eνk−t)

= 0 .

which implies that since the subspace F0 spanned by eE1 , . . . , eE

N , e1, . . . , en′ is k-isotropic,the subspace F1 spanned by eE

1 , . . . , eEN , e1, . . . , en′ and u is so as well. 2

Now it is easy to derive the algebraic Darboux theorem: let ei, eµ | 1 6 i 6 N ,1 ≤ µ ≤ n be a basis of a k-isotropic subspace F complementary to L in W , with dualbasis ei, eµ | 1 6 i 6 N , 1 ≤ mu ≤ n of L⊥ ∼= F ∗, such that ei | 1 6 i 6 N is a basisof the (r − 1)-isotropic subspace E = V ∩F which is complementary to L in V . Choosingany subspace L′ of L complementary to ker ω and taking into account the identity (1.59),we define a basis

ei1... is; µ1... µk−s | 0 6 s 6 r − 1 ,1 6 i1 < . . . < is 6 N

1 6 µ1 < . . . < µk−s 6 n

of L′ byω[

V (ei1... is; µ1... µk−s) = ei1 ∧ . . . ∧ eis ∧ eµ1 ∧ . . . ∧ eµk+1−s .

It is easy to see that the union of this basis with that of F gives a canonical basis of W (ormore precisely, of F ⊕L′, which is a subspace of W complementary to ker ω). Thus we haveproved

1.6 Multilagrangian vector spaces 27

Theorem 1.11 (Darboux theorem for multilagrangian vector spaces)

Every multilagrangian vector space admits a canonical basis.

Once again, the inductive construction of a k-isotropic subspace F complementary to themultilagrangian subspace L, as presented in the proof of Theorem 1.10, provides an iterativeand explicit method to construct multilagrangian bases in a way similar to the well knownGram-Schmidt orthogonalization process.

Chapter 2

Polylagrangian and multilagrangianforms: differential theory

The algebraic concepts introduced in the previous chapter constitute the basis for the de-velopment of a coherent differential theory that will be presented in this chapter. Our mainresult will be a Darboux theorem for polylagrangian and multilagrangian forms which guar-antees that differential forms which are pointwise polylagrangian or multilagrangian admitholonomic canonical bases, or in other words, canonical local coordinates, if and only if theysatisfy the standard integrability condition of being closed.

Regarding terminology, we shall freely use the standard notions of the theory of manifoldsand of fiber bundles (in particular, vector bundles). A typical result that will be usedrepeatedly is the following.

Lemma 2.1 Let E1 and E2 be vector subbundles of a vector bundle E. Then E1 ∩E2 hasconstant dimension if and only if E1 +E2 has constant dimension, and in this case, E1 ∩E2

and E1 + E2 are vector subbundles of E as well.

A similar assertion holds for the intersection and sum of a finite number of vector subbundlesof a given vector bundle.

As a particular case, we have the concept of a distribution on a manifold P , which issimply a vector subbundle D of the tangent bundle TP of P , thus formalizing the idea of afamily of subspaces Dp of the tangent spaces TpP parametrized by the points p of P in sucha way that the subspaces Dp “depend smoothly on p” and “have constant dimension” (i.e.,dim Dp does not depend on p). Such a distribution is involutive if for any two vector fieldsX and Y which are sections of D, their Lie bracket [X, Y ] is again a section of D, and it isintegrable if it arises from a foliation of P , i.e., a decomposition of P into a disjoint unionof submanifolds called leaves such that for any point of P the space Dp is the tangent space

29

30 Polylagrangian and multilagrangian forms: differential theory

to the leaf passing through p. According to the Frobenius theorem, integrable distributionsare involutive and, conversely, involutive distributions are integrable.

One well known example of an involutive distribution is the kernel of a closed differentialform of constant rank. Briefly, the kernel ker ω of a (k+1)-form ω on a manifold P , definedby

kerp ω = ker ωp = up ∈TpP | iupω = 0 , (2.1)

gives rise to a distribution on P as soon as it has constant dimension, i.e., as soon as ω hasconstant rank.1 Using the general formula for the exterior derivative of ω,

dω (X1, . . . , Xk+2)

=k+2∑i=1

(−1)i−1 Xi ·(ω(X1, . . . , Xi, . . . , Xk+2)

)+

∑16i<j6k+2

(−1)i+j ω([Xi, Xj], X1, . . . , Xi, . . . , Xj, . . . , Xk+2)

, (2.2)

we conclude that if dω = 0 and if X1 and X2 take values in ker ω, then [X1, X2] takes valuesin ker ω as well, that is,

dω = 0 =⇒ ker ω involutive . (2.3)

The same procedure works for vector-valued differential forms.

2.1 Cartan calculus for vertical forms

In order to extend the structures studied in the previous chapter (specifically, the symbol)from a purely algebraic setting to the realm of differential geometry, we shall need a vari-ant of Cartan’s calculus, which in its standard formulation deals with differential forms onmanifolds, to handle vertical differential forms on total spaces of fiber bundles.

Let E be a fiber bundle over a base manifold M , with projection π : E → M , andlet T a vector bundle over the same base manifold M , with projection τ : T → M .Consider the pull-back of T to E, which will be denoted by π∗T , and the vertical bundleV E of E: both are vector bundles over E. Then the vector bundle∧r

V ∗E ⊗ π∗T

1Although there exist different conventions regarding the concept of rank (the most common one beingthe codimension of the kernel), the notion of constant rank is unambiguously defined.

2.1 Cartan calculus for vertical forms 31

will be called the bundle of vertical r-forms on E, and its sections are called differentialr-forms or simply r-forms on E, with values or coefficents in π∗T : the space of suchforms will be denoted by Ωr

V (E ; π∗T ). On the other hand, the sections of the vertical bundleVE are called vertical vector fields or simply vertical fields on E: the space of such fieldswill be denoted by XV (E). Obviously, XV (E) and Ωr

V (E ; π∗T ) are (locally finite) modulesover the algebra F(E) of functions on E.

It should be noted that there is a certain amount of “abuse of language” in this termi-nology, since the elements of Ωr

V (E ; π∗T ) are really only equivalence classes of differentialr-forms on E, since Ωr

V (E ; π∗T ) is not a subspace of the space Ωr(E ; π∗T ) of all differentialr-forms on E but rather its quotient space

ΩrV (E ; π∗T ) = Ωr(E ; π∗T )/Ω r

r−1(E ; π∗T )

by the subspace Ω rr−1(E ; π∗T ) of all 1-horizontal differential r-forms on E.

An interesting aspect of this construction is that it is possible to develop a variant ofthe usual Cartan calculus for differential forms on a manifold M in which functions on Mare replaced by functions f on E, vector fields on M are replaced by vertical fields X on Eand differential forms on M are replaced by vertical differential forms α on E taking valuesin π∗T , in such a way that all operations of this calculus are preserved and continue to satisfythe same rules. (See [10, Vol. 1, Probl. 8, p. 313] for the special case where T is a trivialfiber bundle M × R.) These operations are the contraction

i : XV (E)× ΩrV (E ; π∗T ) −→ Ωr−1

V (E ; π∗T )

(X, α) 7−→ iXα,

the vertical Lie derivative

L : XV (E)× ΩrV (E ; π∗T ) −→ Ωr

V (E ; π∗T )

(X, α) 7−→ LXα, (2.4)

and the vertical exterior derivative

dV : ΩrV (E ; π∗T ) −→ Ωr+1

V (E ; π∗T )

α 7−→ dV α, (2.5)

apart from various versions of the exterior product, for instance

∧ : ΩrV (E ; π∗T )× Ωs

V (E ; π∗T ′) −→ Ωr+sV (E ; π∗(T ⊗ T ′))

(α, α′) 7−→ α ∧α′.


The first and the last of these are purely algebraic operations, identical with the usual ones.To define the second and third, we use the same formulas as in the usual case, namely

LXα (X1, . . . , Xr) = X · (α(X1, . . . , Xr)) −r∑

i=1

α(X1, . . . , [X, Xi], . . . , Xr) , (2.6)

and

dV α (X0, . . . , Xr) =r∑

i=0

(−1)i Xi ·(α(X0, . . . , Xi, . . . , Xr)

)+

∑06i<j6r

(−1)i+j α([Xi, Xj], X0, . . . , Xi, . . . , Xj, . . . , Xr), (2.7)

where X0, X1, . . . , Xr ∈ XV (E) , which makes sense since V E is an involutive distributionon E (i.e., XV (E) is a subalgebra of X(E) with respect to the Lie bracket), provided wecorrectly define the vertical directional derivative

XV (E)× Γ(π∗T ) −→ Γ(π∗T )

(X, ϕ) 7−→ X · ϕ. (2.8)

as an R-bilinear operator which is F(E)-linear in the first entry and satisfies the Leibniz rulein the second entry,

X · (fϕ) = (X · f) ϕ + f(X · ϕ) . (2.9)

Explicitly, for X ∈ XV (E) and ϕ ∈ Γ(π∗T ), X·ϕ ∈ Γ(π∗T ) is defined as the standard direc-tional derivative of vector valued functions along the fibers, that is, for any point m in M ,(X · ϕ)

∣∣Em

∈ C∞(Em, Tm) is given in terms of X∣∣Em

∈ X(Em) and ϕ∣∣Em

∈ C∞(Em, Tm) by

(X · ϕ)∣∣Em

= X∣∣Em· ϕ∣∣Em

. (2.10)

Since the Lie bracket is natural under restriction to submanifolds, we have

X ·(Y · ϕ

)− Y ·

(X · ϕ

)= [X, Y ]· ϕ for X, Y ∈ XV (E), ϕ ∈ Γ(π∗T ) ,

which implies that d2V = 0. On the other hand, sections ϕ of π∗T obtained from sections t

of T by composing with π are constant along the fibers and hence their vertical directionalderivative vanishes:

X · (t π) = 0 for X ∈ XV (E), t ∈ Γ(T ) . (2.11)

In the same way, substituting T by T ∗, we get

X · (t∗ π) = 0 for X ∈ XV (E), t∗ ∈ Γ(T ∗) , (2.12)

2.2 Polylagrangian manifolds 33

which means that

X · 〈t∗ π , ϕ〉 = 〈t∗ π , (X · ϕ)〉 for X ∈ XV (E), ϕ ∈ Γ(π∗T ) . (2.13)

More generally, given a π∗T -valued vertical r-form ω on P and a section t∗ of the dual vectorbundle T ∗ of T , we define the projection of ω along t∗ to be the ordinary vertical r-form ω

t∗

on P given by 2

ωt∗(p) = 〈t∗(π(p)), ω(p)〉 for p ∈P , (2.14)

and obtaindV ωt∗ =

(dV ω

)t∗

. (2.15)

Hence, if ω is closed, ωt∗

will be closed as well. Moreover, the kernel ker ω of ω and thekernel ker ω

t∗of ω

t∗, defined by

kerp ω = ker ωp = up ∈VpP | iupω = 0 for p ∈P , (2.16)

andkerp ωt∗ = ker(ωt∗)p = up ∈VpP | iup

ωt∗ = 0 for p ∈P , (2.17)

satisfy ker ω ⊂ ker ωt∗⊂V P and define distributions on P of P as long as they have constant

dimension (in the case of ker ω , this is equivalent to the condition that ω has constant rank),and applying the same procedure as before, we conclude that

dV ω = 0 =⇒ ker ω and ker ωt∗ are involutive . (2.18)

2.2 Polylagrangian manifolds

In this section and the next, we shall transfer the poly- and multilagrangian structuresintroduced in the previous chapter from the algebraic to the differential context by addingthe appropriate integrability condition.

Definition 2.1 A polylagrangian manifold is a manifold P equipped with a (k + 1)-form ω taking values in a fixed n-dimensional vector space T , called the polylagrangianform and said to be of rank N , such that ω is closed,

dω = 0 , (2.19)

and admits a distribution L on P , called the polylagrangian distribution, such that atevery point p of P , ωp is a polylagrangian form of rank N on TpP with polylagrangiansubspace Lp.When k = 1, we say that P is a polysymplectic manifold and ω is a polysymplecticform.

2Here, it is important that t∗ be a section of T ∗ and not of π∗T ∗.


Note that according to Theorem 1.2 and equation (1.47), L has constant dimension if andonly if ker ω has constant dimension, that is, ω has constant rank.

As in the purely algebraic context, we associate with each linear form t∗ ∈ T ∗ on theauxiliary space T an ordinary (k +1)-form ωt∗ on P called the projection of ω along t∗ anddefined by

ωt∗ = 〈t∗, ω〉 , (2.20)

so taking a basis e1, . . . , en of T with dual basis e1, . . . , en of T ∗, and defining theordinary (k + 1)-forms

ωa = ωea (1 6 a 6 n) , (2.21)

we obtain, as in the purely algebraic context,

ker ω =⋂

t∗∈ T ∗

ker ωt∗ =n⋂

a=1

ker ωa . (2.22)

When n ≡ dim T > 2, we can use Theorem 1.4 to conclude that

L =∑

t∗∈ T ∗\0

ker ωt∗ . (2.23)

In this case, it is sufficient to suppose that ω is a T -valued (k + 1)-form of constant ranksuch that at every point p of P , Lp is the polylagrangian subspace of TpP with respect to ωp,since the dimension formulas (1.46) and (1.51) then guarantee that, for 1 6 a 6 n, ker ωa

has constant dimension and hence defines a distribution on P ; therefore, L is a distributionon P as well. Moreover, for 1 6 a 6 n, Ka has constant dimension and can be chosento be a distribution on P . It should be emphasized, however, that this argument failsif n ≡ dim T = 1, that is, for ordinary forms – notably in the symplectic case, where alagrangian distribution is far from unique and a mere collection of lagrangian subspacesneed not even be differentiable.

When n ≡ dim T > 3, we can apply Theorem 1.4 to show that the polylagrangiandistribution is necessarily involutive!

Theorem 2.1 Let P be a polylagrangian manifold with polylagrangian (k+1)-form ω takingvalues in a fixed vector space T of dimension n > 3. Then the polylagrangian distribution Lis involutive.

Proof. With the same notation as before, suppose that X and Y are vector fields whichare sections of L. Using the decomposition

L = K0 ⊕K1 ⊕ . . .⊕Kn ,

2.3 Poly- and multilagrangian fiber bundles 35

with K0 = ker ω (see equation (1.48)), we can decompose X and Y according to

X =n∑

a=0

Xa , Y =n∑

b=0

Yb ,

where Xa and Yb are sections of Ka and Kb, respectively. Using that n > 3, we can foreach value of a and b find a value c 6= 0 such that c 6= a and c 6= b ; then Ka ⊂ ker ωc andKb ⊂ ker ωc . Since ωc is closed and has constant rank, ker ωc is involutive. Therefore thevector field [Xa, Yb] is a section of ker ωc ⊂L. 2

There are polylagrangian manifolds whose polylagrangian distribution is not involutive.According to Theorem 2.1, this can only occur when n = 1 or n = 2. An interestingexample of this kind is the following.

Example 2.1 Let P = S3 = SU(2), T = R2 and define α1, α2, α3 to be the threecomponents of the left invariant Maurer-Cartan form on P with respect to the standardbasis of R3, which are dual to the three left invariant vector fields ξ1, ξ2, ξ3 generated by thisbasis. Define

ω(1) = α3 ∧ α1 , ω(2) = α3 ∧ α2 , ω(3) = α1 ∧ α2 .

It follows immediately from the Maurer-Cartan structure equations that each of these 2-formsis closed. Thus omitting ω(3), say, we see that

ω = ω(1)e1 + ω(2)e2

is a non-degenerate polysymplectic form of rank 1 on P whose polylagrangian distributionis spanned by ξ1 (which generates the kernel of ω(2)) and ξ2 (which generates the kernelof ω(1)), and this is certainly not involutive.

This implies that for n = 1 or n = 2, the condition of integrability of the polylagrangiandistribution L needs to be imposed separately, whenever necessary or convenient.

2.3 Poly- and multilagrangian fiber bundles

Passing from manifolds to fiber bundles, suppose that P is a bundle over a base manifold Mwith projection π : P → M and consider the tangent mapping Tπ : TP → TM , whose


kernel is a vector subbundle V P of the tangent bundle TP of P , called the vertical bundle,giving rise to the following exact sequence of vector bundles over P :

0 −→ V P −→ TPTπ−→ π∗TM −→ 0 . (2.24)

This will play the same role as the exact sequence (1.3) of vector spaces in the previouschapter and is the basis for the notions of horizontality that will appear in this chapter.

To begin with, we introduce the notion of a polylagrangian fiber bundle, which formalizesthe idea of a “family of polylagrangian manifolds smoothly parametrized by the points of abase manifold M”. To do so, we shall need the concepts introduced in the first section ofthis chapter.

Definition 2.2 A polylagrangian fiber bundle is a fiber bundle P over an n-dimensional manifold M equipped with a vertical (k + 1)-form ω on the total space P takingvalues in a fixed n-dimensional vector bundle T over the same manifold M , called the poly-lagrangian form along the fibers of P , or simply polylagrangian form, and said tobe of rank N , such that ω is vertically closed,

dV ω = 0 , (2.25)

and admits a distribution L on P contained in the vertical bundle V P of P , called thepolylagrangian distribution, such that at every point p of P , ωp is a polylagrangian formof rank N on VpP with polylagrangian subspace Lp.When k = 1, we say that P is a polysymplectic fiber bundle and ω is a polysymplecticform along the fibers of P , or simply polysymplectic form.

If P is a polylagrangian fiber bundle over a manifold M with polylagrangian form ω takingvalues in a vector bundle T over M , then for any point m in M , the fiber Pm of P over m willbe a polylagrangian manifold with polylagrangian form ωm, the restriction of ω to Pm, takingvalues in the vector space Tm, the fiber of T over m, and whose polylagrangian distributionLm is the restriction of the polylagrangian distribution L to Pm. The main point of the abovedefinition is to guarantee regularity of this family of polylagrangian manifolds, parametrizedby the points m in M , along M , that is, transversally to the fibers. Obviously, the involutivitytheorem (Theorem 2.1) remains valid in this context.

In the same way, we define the notion of a multilagrangian fiber bundle:

Definition 2.3 A multilagrangian fiber bundle is a fiber bundle P over an n-dimensional manifold M equipped with a (k + 1− r)-horizontal (k + 1)-form ω on the totalspace P , where 1 6 r 6 k and k + 1− r 6 n, called the multilagrangian form and saidto be of rank N and horizontal degree k + 1− r, such that ω is closed,

dω = 0 , (2.26)

2.4 The symbol 37

and admits a distribution L on P , called the multilagrangian distribution, such that atevery point p of P , ωp is a multilagrangian form of rank N and horizontal degree k + 1− ron TpP with multilagrangian subspace Lp.When k = n, r = 2 and ω is non-degenerate, we say that P is a multisymplectic fiberbundle and ω is a multisymplectic form.

2.4 The symbol

In what follows we want to show how the close relation between multi- and polylagrangianstructures found in the previous chapter can be extended from the purely algebraic contextto that of differential geometry: the link is once again provided by the concept of symbol,which is now a prescription that to each multilagrangian form on a fiber bundle associates apolylagrangian form on that fiber bundle taking values in an appropriate bundle of differentialforms on the base manifold. To this end, we shall once again need the concepts introducedin the first section of this chapter.

Thus let us suppose, as in the previous section, that P is a fiber bundle over an n-dimensional base manifold M and let us use the exact sequence (2.24). We also suppose,again as in the previous section, that ω is a (k + 1− r)-horizontal (k + 1)-form on P , where1 6 r 6 k and k + 1− r 6 n:

ω ∈ Ω k+1r

(P)

= Γ(∧k+1

r T ∗P)

. (2.27)

Explicitly, this means that contraction of ω with more than r vertical fields on P gives zero:

iX1. . . iXr+1

ω = 0 for X1, . . . , Xr+1 ∈ XV (P ) . (2.28)

Then the symbol ω of ω, defined pointwise (i.e., such that at every point p of P , ωp is thesymbol of ωp in the sense of the previous chapter), will be a vertical r-form on P taking

values in the vector bundle π∗(∧k+1−r

T ∗M):

ω ∈ Ω rV

(P , π∗

(∧k+1−rT ∗M

))= Γ

(∧rV ∗P ⊗ π∗

(∧k+1−rT ∗M

)). (2.29)

Using the canonical isomorphism π∗(∧k+1−r

T ∗M) ∼= ∧k+1−r

0 T ∗P of vector bundles over Pas an identification, we have

ω(X1, . . . , Xr) = iX1. . . iXr

ω for X1, . . . , Xr ∈ XV (P ) . (2.30)

More explicitly, using the horizontal lift

X(M) −→ X(P )

ξ 7−→ ξH


of vector fields induced by some fixed connection on P , we have

ω(X1, . . . , Xr) ·(ξ1 π, . . . , ξk+1−r

π)

= ω(X1, . . . , Xr, ξH1 , . . . , ξH

k+1−r)

for X1, . . . , Xr ∈ XV (P ), ξ1, . . . , ξk+1−r ∈ X(M) ,(2.31)

where it should be noted that although composition of vector fields on M with the projectionπ provides only a subspace of the vector space of all sections of the pull-back π∗TM of TMby π, this formula is sufficient to fix the value of ω at each point p of P .

Theorem 2.2 With the same notations as above, suppose that the form ω satisfies

dω ∈ Ω k+2r

(P)

= Γ(∧ k+2

r T ∗P)

,

i.e., dω is (k + 2− r)-horizontal. Then the form ω is vertically closed:

dV ω = 0 .

In particular, for ω to be vertically closed, it is sufficient but not necessary that ω be closed.

Proof. Let X0, . . . , Xr be vertical fields on P and ξ1, . . . , ξk+1−r be vector fields on M ,and denote the horizontal lifts of the latter with respect to some fixed connection on P byξH1 , . . . , ξH

k+1−r, respectively. Then we compute(dV ω

(X0, . . . , Xr

))(ξ1 π, . . . , ξk+1−r π

)=

(r∑

i=0

(−1)i Xi ·(ω(X0, . . . , Xi, . . . , Xr)

)+

∑06i<j6r

(−1)i+j ω([Xi, Xj], X0, . . . , Xi, . . . , Xj, . . . , Xr)

)(ξ1 π, . . . , ξk+1−r π

)=

r∑i=0

(−1)i Xi ·(ω(X0, . . . , Xi, . . . , Xr)

(ξ1 π, . . . , ξk+1−r π

))+

∑06i<j6r

(−1)i+j ω([Xi, Xj], X0, . . . , Xi, . . . , Xj, . . . , Xr)(ξ1 π, . . . , ξk+1−r π

)=

r∑i=0

(−1)i Xi · ω(X0, . . . , Xi, . . . , Xr, ξH1 , . . . , ξH

k+1−r)

+k+1−r∑

j=1

(−1)r+j ξHj · ω(X0, . . . , Xr, ξ

H1 , . . . , ξH

j , . . . , ξHk+1−r)

2.4 The symbol 39

+∑

06i<j6r

(−1)i+j ω([Xi, Xj], X0, . . . , Xi, . . . , Xj, . . . , Xr, ξH1 , . . . , ξH

k+1−r)

+r∑

i=0

k+1−r∑j=1

(−1)i ω([Xi, ξHj ], X0, . . . , Xi, . . . , Xr, ξ

H1 , . . . , ξH

j , . . . , ξHk+1−r)

+∑

16i<j6k+1−r

(−1)i+j ω([ξHi , ξH

j ], X0, . . . , Xr, ξH1 , . . . , ξH

i , . . . , ξHj , . . . , ξH

k+1−r)

= dω(X0, . . . , Xr, ξ

H1 , . . . , ξH

k+1−r

),

where in the second equality we use equation (2.13) with ϕ = ω(X0, . . . , Xi, . . . , Xr) andt∗ = ξ1 ∧ . . . ∧ ξk+1−r and in the third equality we have added terms which vanish due to thecondition that ω should be (k +1− r)-horizontal, since the Lie brackets [Xi, ξ

Hj ] are vertical.

(More generally, if X is a projectable vector field on P and π∗X its projection to M , then thepairs (X, π∗X) and (ξH , ξ) are π-related, so the pair ([X, ξH ], [π∗X, ξ]) will also be π-related.But since X is vertical we have π∗X = 0.) 2

From Theorems 1.7 and 2.2, it follows that

Theorem 2.3 Let P be a fiber bundle over an n-dimensional manifold M , with projectionπ : P → M , let ω be a (k + 1 − r)-horizontal (k + 1)-form on P , where 1 6 r 6 k andk + 1− r 6 n, and let ω be its symbol, which is a vertical r-form on P taking values in the

bundle of (k + 1 − r)-forms on M (or more exactly, in π∗(∧k+1−r

T ∗M)). Suppose that ω

is multilagrangian, with multilagrangian distribution L. Then ω will be polylagrangian, withpolylagrangian distribution L. If ω is multisymplectic, then ω will be polysymplectic.

Regarding the question of involutivity of the poly- or multilagrangian distribution on fiberbundles, we can guarantee that

L is involutive if n > 3, in the polylagrangian case , (2.32)

and

L is involutive if(

nk+1−r

)> 3, in the multilagrangian case , (2.33)

the argument being exactly the same as that in the proof of Theorem 2.1, that is, wewrite L as a finite sum of involutive distributions which are kernels of projected forms ωa.For multilagrangian fibrer bundles, it is interesting to spell out more explicitly under whatcircumstances L may fail to be involutive:


1. n = k + 1− r : this includes the symplectic case (r = 2) with

ω ∈ Ωn+22

(P)

, ω ∈ Ω 2V

(P , π∗

(∧nT ∗M

)),

representing, under the additional hypothesis that ker ω = 0, a “family of symplecticforms smoothly parametrized by the points of a base manifold M” of dimension n.Here, it is not difficult to construct examples of lagrangian distributions which are notinvolutive.

2. n > k +1− r : the only possibility which is compatible with the condition(

nk+1−r

)= 2

is n = 2 and k = r : this includes the multisymplectic case over a two-dimensionalbase manifold M (n = k = r = 2). An explicit example of this situation can be givenby making use of the following construction, which is an adaptation of Example 2.1.

Example 2.2 Let M be a two-dimensional manifold admitting closed 1-forms θ1 and θ2

such that θ1∧θ2 is a volume form on M (for instance, the two-dimensional torus or anyof its coverings), and let P be the total space of a principal bundle over M with struc-ture group U(2). Given any connection form A ∈Ω1(P, u(2)) on P with curvature formF ∈Ω2(P, u(2)), and using the basis i1 , σ1/2i , σ2/2i , σ3/2i of u(2), where

σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

),

are the Pauli matrices, so that

A = A0 i1 + Aa σa

2i, F = F 0 i1 + F a σa

2i,

and

F 0 = dA0 , F a = dAa + 12εabcAb ∧ Ac ,

we have that

ω = A3 ∧ A1 ∧ π∗θ1 + A3 ∧ A2 ∧ π∗θ2 − A0 ∧ π∗θ1 ∧ π∗θ2

is a multisymplectic form of rank 1 on P whose multilagrangian distribution L is spannedby the fundamental vector fields on P associated with the generators σ1/2i, σ2/2i and i1,which is not involutive. Indeed, with L defined in this way and observing that the verticalbundle V P is spanned by the fundamental vector fields on P associated with the generatorsσ1/2i, σ2/2i, σ3/2i and i1, it becomes obvious that ω vanishes when contracted with at leastthree vector fields taking values in V P or two vector fields taking values in L, so all that

2.5 The Darboux Theorem 41

remains to be shown is that ω is closed. But this is a straightforward computation, since

dω = dA3 ∧ A1 ∧ π∗θ1 − A3 ∧ dA1 ∧ π∗θ1 + A3 ∧ A1 ∧ π∗(dθ1)

+ dA3 ∧ A2 ∧ π∗θ2 − A3 ∧ dA2 ∧ π∗θ2 + A3 ∧ A2 ∧ π∗(dθ2)

− dA0 ∧ π∗θ1 ∧ π∗θ2 + A0 ∧ π∗(dθ1) ∧ π∗θ2 − A0 ∧ π∗θ1 ∧ π∗(dθ2)

= F 3 ∧ A1 ∧ π∗θ1 − A3 ∧ F 1 ∧ π∗θ1 + F 3 ∧ A2 ∧ π∗θ2 − A3 ∧ F 2 ∧ π∗θ2

− F 0 ∧ π∗θ1 ∧ π∗θ2 ,

and since each of the terms in the last equation is at least 3-horizontal and hence mustvanish because M is two-dimensional. Note that the symbol of this multisymplectic form isthe polysymplectic form

ω =(A3|V P ∧ A1|V P

)⊗ θ1 +

(A3|V P ∧ A2|V P

)⊗ θ2 ,

which is essentially the polysymplectic form of Example 2.1 (trivially extended from SU(2)to U(2)).

2.5 The Darboux Theorem

Now we are able to prove the Darboux theorem for poly- and multilagrangian forms. Here,the specific algebraic structure of poly- and multilagrangian subspaces identified in the previ-ous chapter turns out to be crucial, in the sense that this central theorem can, in all cases, beproved by appropriately adapting the procedure used to prove the classical Darboux theoremfor symplectic forms. (See, for instance, Ref. [1]).

Theorem 2.4 (Darboux theorem for polylagrangian manifolds) Let P be a poly-lagrangian manifold with polylagrangian (k + 1)-form ω of rank N taking values in a fixedn-dimensional vector space T and with polylagrangian distribution L, which is assumed to beinvolutive (recall this is automatic if n > 3), and let ea | 1 6 a 6 n be a basis of T . Thenaround any point of P , there exists a system of local coordinates (qi, p a

i1... ik, rκ) (1 6 a 6 n,

1 6 i 6 N , 1 6 i1 < . . . < ik 6 N , 1 6 κ 6 dim ker ω), called Darboux coordinates orcanonical coordinates, such that

ω = 1k!

dpai1... ik

∧ dqi1 ∧ . . . ∧ dqik ⊗ ea , (2.34)

and such that (locally) L is spanned by the vector fields ∂/∂pai1... ik

and ∂/∂rκ while ker ωis spanned by the vector fields ∂/∂rκ.


Theorem 2.5 (Darboux theorem for polylagrangian fiber bundles) Let P be apolylagrangian fiber bundle over an n-dimensional manifold M with polylagrangian (k + 1)-form ω of rank N taking values in a fixed n-dimensional vector bundle T over the samemanifold M and with polylagrangian distribution L, which is assumed to be involutive (recallthis is automatic if n > 3), and let ea | 1 6 a 6 n be a basis of local sections of T .Then around any point of P (over the domain of the given basis of local sections), thereexists a system of local coordinates (xµ, qi, pa

i1... ir−1, rκ) (1 6 µ 6 n, 1 6 a 6 n, 1 6 i 6 N ,

1 6 i1 < . . . < ir−1 6 N , 1 6 κ 6 dim ker ω), called Darboux coordinates or canonicalcoordinates, such that

ω = 1k!

dpai1... ik

∧ dqi1 ∧ . . . ∧ dqik ⊗ ea , (2.35)

and such that (locally) L is spanned by the vector fields ∂/∂pai1... ik

and ∂/∂rκ while ker ωis spanned by the vector fields ∂/∂rκ.

Theorem 2.6 (Multilagrangian Darboux theorem) Let P be a multilagrangian fiberbundle over an n-dimensional manifold M with multilagrangian (k+1)-form ω of rank N andhorizontal degree k+1−r, where 1 6 r 6 k and k+1−r 6 n, and with multilagrangian dis-tribution L, which is assumed to be involutive (recall this is automatic if

(n

k+1−r

)> 3). Then

around any point of P , there exists a system of local coordinates (xµ, qi, p i1... is; µ1... µk−s, rκ)

(0 6 s 6 r − 1, 1 6 µ 6 n, 1 6 i 6 N , 1 6 i1 < . . . < is 6 N , 1 6 µ1 < . . . < µk−s 6 N ,1 6 κ 6 dim ker ω), called Darboux coordinates or canonical coordinates, such that

ω =r−1∑s=0

1s!

1(k−s)!

dpi1... is; µ1... µk−s∧ dqi1 ∧ . . . ∧ dqis ∧ dxµ1 ∧ . . . ∧ dxµk−s , (2.36)

and such that (locally) L is spanned by the vector fields ∂/∂pi1... is; µ1... µk−sand ∂/∂rκ while

ker ω is spanned by the vector fields ∂/∂rκ. In these coordinates, its symbol is given by

ω = 1(r−1)!

1(k+1−r)!

(dpi1... ir−1; µ1... µk+1−r

∧ dqi1 ∧ . . . ∧ dqir−1)⊗(dxµ1 ∧ . . . ∧ dxµk+1−r

). (2.37)

Proof: For the sake of simplicity, we concentrate on the multilagrangian case: the prooffor the other two cases is entirely analogous, requiring only small and rather obvious modi-fications.

Due to the local character of this theorem and since the kernel of ω, the multilagrangiansubbundle L and the vertical bundle VP are all involutive, with ker ω ⊂L ⊂VP , we can with-out loss of generality work in a local chart of the manifold P around the chosen referencepoint in which the corresponding foliations are “straightened out”, so we may assume thatP ∼= Rn ⊕ RN ⊕ L0 ⊕K0 with VP ∼= RN ⊕ L0 ⊕K0, L ∼= L0 ⊕K0 and ker ω ∼= K0 withfixed subspaces L0 and K0 and such that the aforementioned reference point corresponds to

2.5 The Darboux Theorem 43

the origin. We also take ω0 to be the constant multilagrangian form, with multilagrangiandistribution L, obtained by spreading ω(0), the value of the multilagrangian form ω at theorigin, all over the space P ; then the existence of canonical coordinates for ω0, in the formgiven by equation (2.36), is already guaranteed by the algebraic Darboux theorem of theprevious chapter (Theorem 1.11).

Now consider the family of (k + 1)-forms given by ωt = ω0 + t(ω − ω0), for every t ∈R.Obviously, ωt(0) = ω0 for every t ∈R, which is non-degenerate on K ′

0 = Rn ⊕ RN ⊕ L0 (acomplement of K0 in P ). Since non-degeneracy is an open condition, and using a compactnessargument with respect to the parameter t, we conclude that there is an open neighborhoodof 0 such that, for all t satisfying 0 6 t 6 1 and all points p in this neighborhood, ωt(p)is non-degenerate on K ′

0 = Rn ⊕ RN ⊕ L0 , that is, its kernel equals K0. Moreover, forall t satisfying 0 6 t 6 1 and all points p in this neighborhood, the subspace L0, beingisotropic for ω0 as well as for ω(p), is also isotropic for ωt(p) and, since it contains the kernelof ωt(p) and has the right dimension as given by equation (1.62), is even multilagrangianfor ωt(p), according to Theorem 1.6. On the other hand, we have dω0 = 0 (trivially) anddω = 0 (by hypothesis), so we can apply an appropriate version of the Poincare lemma (seethe Appendix) to prove, in some open neighborhood of the point 0 in P (contained in theprevious one), existence of a k-form α satisfying dα = ω0 − ω and α[(L) = 0. Now take Xt

to be the unique time dependent vector field on P taking values in L03 defined by

iXtωt = α .

Let Ft ≡ F(0,t) be its flux beginning at 0, once again defined, for 0 6 t 6 1, in some openneighborhood of the point 0 in P (contained in the previous one). Then it follows that

d

ds

∣∣∣∣s=t

F ∗s ωs = F ∗

t

(d

ds

∣∣∣∣s=t

ωs

)+

d

ds

∣∣∣∣s=t

F ∗s ωt

= F ∗t

(ω − ω0 + LXt

ωt

)= F ∗

t

(ω − ω0 + d(iXt

ωt))

= F ∗t

(ω − ω0 + dα

)= 0

Therefore, F1 is the desired coordinate transformation, since F ∗1 ω = F ∗

1 ω1 = F ∗0 ω0 = ω0.

(For additional information, see [1].) 2

3It is at thios point that we make essential use of the hypothesis that L0 is multilagrangian and not justisotropic or even maxiaml isotropic (with respect to ωt(p), in this case).

Chapter 3

Poly- and multisymplectic structures

In this chapter, we turn to a more specific study of poly- and multisymplectic forms. Ourmain goal is to provide, at least for these important classes of poly- and multilagrangianforms, respectively, a more substantial motivation for their definition. After all, our defi-nition, based exclusively on the existence of a special type of maximal isotropic subspace(called a poly- or multilagrangian subspace, respectively), constitutes the central point ofthis work and should be compared with other proposals for defining the same concepts thatcan be found in the literature, which calls for a critical assessment.

A first attempt to define the notion of a multisymplectic manifold (P, ω) and prove aDarboux theorem is due to Martin [19]. The idea of postulating the existence of a “big”isotropic subspace already appeared in this paper, but the precise mathematical meaning ofthat condition remained unclear, partly because the criterion for being “big” was formulatedmerely as a numerical condition on the dimension. Soon it was noticed that the resultingformula for the total dimension (which follows from equation (1.45) or (1.46) by setting n=1and ker ω = 0), namely

dim P = N +

(N

k

),

is inconsistent with the basic example from physics, because the dimension of the multiphasespaces (P, ω) that appear in the covariant hamiltonian formalism of classical field theory isgiven by a different formula (which follows from equation (1.61) or (1.62) by setting k =n,r=2 and ker ω = 0), namely

dim P = (N + 1)(n + 1) ,

which can also be obtained by a simple count of local coordinates: in the notation alreadyemployed in the introduction, we have n space-time coordinates xµ, N position variablesqi, nN multimomentum variables p µ

i and one energy variable p . Therefore, and takinginto account that the term “multisymplectic” is already occupied in the literature at least

45

46 Poly- and multisymplectic structures

since the beginning of the 1970s [13–15], we found it inadequate to use the same term forthe structure studied by Martin in Ref. [19] and thus propose to replace it by the term“polylagrangian” – a convention that we have adopted in this work.

In later articles, such as [3] and [17], the necessity of modifying and extending the for-malism by including horizontality conditions was already clearly realized, but so far, at leastto our knowledge, there exists no comprehensive treatment of the subject, which would haveto include complete proofs of the main statements.

There are also approaches emphasizing other aspects that, in our opinion, turn out to beof little relevance. One example that we shall discuss in more detail are the hypotheses onthe symmetric algebra built from exterior products between the members of a multiplet of 2-forms adopted in Ref. [11]: as we shall demonstrate, these conditions, apart from the apparentimpossibility to extend them naturally from the polysymplectic to the multisymplectic case,are both insufficient and superfluous to achieve the main objective (namely the formulationand proof of an appropriate version of the algebraic Darboux theorem). But in spite of thesedrawbacks, Ref. [11] has played an important role in the development of our approach.

Regarding the relation between polysymplectic and multisymplectic forms, it is interest-ing to note that the latter are almost entirely determined by the former, by means of whatwe call the symbol, since this is a polysymplectic form whose kernel, as we shall see, is atmost one-dimensional. What parametrizes this kernel is a variable which in classical fieldtheory corresponds to energy.

3.1 Polysymplectic vector spaces

According to Definition 1.1, a 2-form ω on a vector space V taking values in another vectorspace T of dimension n is said to be polysymplectic of rank N if there exists a polylagrangiansubspace L of V of codimension N , characterized by the property that

ω[(L)

= L⊥ ⊗ T . (3.1)

In this case, it is obvious that, according to equation (1.45) or (1.46),

dim L = dim ker ω + nN , (3.2)

and hence

dim V = dim ker ω +(n + 1

)N . (3.3)

In particular, the dimension of a vector space carrying a non-degenerate polysymplecticform taking values in an n-dimensional vector space T has to be a multiple of n + 1 :

3.1 Polysymplectic vector spaces 47

this generalizes a well known property of symplectic forms, which are included as the specialcase n = 1 (ordinary forms).1 More generally, we have

rk(ω) = N =1

n + 1dim supp ω , (3.4)

which constitutes another example of the phenomenon already mentioned in the first chapter,according to which it may be convenient to use the term “rank” of a form for an expressionthat is numerically different from the dimension of its support but uniquely determined by it.

When n > 2, we have the following criterion to decide whether a 2-form is polysymplectic:

Proposition 3.1 Let V and T be finite-dimensional vector spaces, with n ≡ dim T > 2,and let ω be a T -valued 2-form on V . Consider the subspace L of V spanned by the kernelsof all the projections ωt∗ (t∗ ∈ T ∗ \ 0) of ω:

L =∑

t∗ ∈ T ∗\0

ker ωt∗ .

For the form ω to be polysymplectic and L to be the pertinent polylagrangian subspace, thefollowing conditions are necessary and sufficient:

• L is isotropic;

• dim L = (dim ker ω + n dim V )/(n + 1).

Proof. This is an immediate corollary of Theorems 1.3 and 1.4 . 2

To formulate the algebraic Darboux theorem in the specific context of polysymplecticforms, we begin by adapting Definition 1.5 to this context: explicitly, a (polysymplectic)Darboux basis or (polysymplectic) canonical basis is a basis ei, e

ja | 1 6 a 6 n ,

1 6 i, j 6 N of a subspace of V complementary to ker ω, with dual basis ei, eaj |

1 6 a 6 n , 1 6 i, j 6 N of the subspace supp ω = (ker ω)⊥ of V ∗, together with abasis ea | 1 6 a 6 n of T , with dual basis ea | 1 6 a 6 n of T ∗, such that

ω =(ea

i ∧ ei)⊗ ea , (3.5)

or in terms of the projected forms ωa defined according to equations (1.29) and (1.30),

ωa = eai ∧ ei (1 6 a 6 n) . (3.6)

1Note that for n=1, the definition of a non-degenerate polysymplectic form reduces to that of a symplecticform, since in this case, the existence of a lagrangian subspace L can be derived as a theorem. The maindifference between this case and those where n>1 is that here, L is not unique.


Obviously, a 2-form ω ∈∧2

V ∗ ⊗ T that admits such a canonical basis is a polysymplecticform, whose polylagrangian subspace L is the direct sum of the kernel of ω with the (nN)-dimensional subspace spanned by the vectors ei

a (1 6 a 6 n, 1 6 i 6 N). It is interestingto note that, on the other hand, the direct sum of the kernel of ω with the N -dimensionalsubspace spanned by the vectors ei (1 6 i 6 N) is a maximal isotropic subspace of V whichis not polylagrangian.

Conversely, as a corollary of Theorem 1.9, we have

Theorem 3.1 (Darboux theorem for polysymplectic vector spaces)

Every polysymplectic vector space admits a canonical basis.

Although this theorem has already been proved in the first chapter, we want to presentan alternative proof which is strictly parallel to the proof of the corresponding theorem forsymplectic forms and for symmetric bilinear forms, based on an inductive “Gram-Schmidt”type process that can also be used to construct, inductively and explicitly, an isotropicsubspace complementary to the polylagrangian subspace L (see, for instance, [1, Prop. 3.1.2,pp. 162-164]).

Proof. Denoting the polylagrangian subspace of V by L, as always, and fixing a subspaceL′ of L complementary to ker ω, we proceed by induction on N = dim V − dim L. To thisend, we choose arbitrarily a 1-form eN ∈ L⊥ \ 0 and a vector eN ∈ V \ L such that

〈eN , eN〉 = 1 ,

and using the property (3.1), define the vectors eNa ∈ L′ (1 6 a 6 n) by 2

ω[(eNa ) = − eN ⊗ ea ,

or in terms of the projected forms

(ωb)[(eNa ) = − δb

a eN ,

and the 1-forms eaN ∈ V ∗ (1 6 a 6 n) by

eaN = (ωa)[(eN) .

Thus we have

〈eN , eN1 〉 = −ω1(eN

1 , eN1 ) = 0 . . . 〈eN , eN

n 〉 = −ωn(eNn , eN

n ) = 0

andωb(eN , eN

a ) = δba .

2Note that this determines the eNa uniquely in terms of eN .


Using that L is isotropic with respect to ω and hence also with respect to each of theprojected forms, we get

ωc(eNa , eN

b ) = 0 .

This implies that we can decompose the space V into the direct sum

V = VN ⊕ V ωN

of the subspace VN spanned by the vectors eN and eNa (1 6 a 6 n), of dimension n + 1,

and its simultaneous orthogonal complement V ωN , which is the intersection of the n different

orthogonal complements of VN with respect to each of the projected forms. What is more:the formula

P ωN(v) = v − 〈eN , v〉 eN − ωa(eN , v) eN

a

provides an explicit definition of the projector P ωN onto V ω

N along VN . [To prove this assertion,note that

P ωN(eN) = eN − 〈eN , eN〉 eN − ωa(eN , eN) eN

a = 0 ,

P ωN(eN

a ) = eNa − 〈eN , eN

a 〉 eN − ωb(eN , eNa ) eN

b = 0 ,

and that, for every v ∈V ,

ωc(P ωN(v), eN) = ωc(v, eN) − 〈eN , v〉ωc(eN , eN) − ωb(eN , v) ωc(eN

b , eN) = 0 ,

ωc(P ωN(v), eN

a ) = ωc(v, eNa ) − 〈eN , v〉ωc(eN , eN

a ) − ωb(eN , v) ωc(eNb , eN

a ) = 0 ,

showing that P ωN vanishes on VN and maps the entire space V to the subspace V ω

N . Usingagain the definition of P ω

N , we also conclude that ker P ωN = VN , since a linear independence

argument shows that P ωN cannot vanish on any vector that does not belong to the subspace

spanned by the vectors eN and eNa (1 6 a 6 n), and that P ω

N is the identity on V ωN , since

using that 〈eN , v〉 = −ωa(eNa , v) for any fixed value of a (no sum) shows that the 1-form

eN vanishes on V ωN . Therefore, P ω

N is a projector (i.e., satisfies (P ωN)2 = P ω

N) with kernel VN

and image V ωN .] Passing to the dual, this direct decomposition of V induces a corresponding

direct decompositionV ∗ = V ∗

N ⊕ (V ωN )∗

of V ∗, where the dual of each subspace is naturally identified with the annihilator of theother:

V ∗N = (V ω

N )⊥ , (V ωN )∗ = V ⊥

N .

Note that under these decompositions L becomes the direct sum of the intersection L ∩V ωN

and the subspace spanned by the vectors eNa (1 6 a 6 n) while L⊥ becomes the direct sum

of the intersection L⊥ ∩V ⊥N and the one-dimensional subspace spanned by the 1-form eN .

Thus it becomes clear that we can repeat the same process for the restriction of the formω to the subspace V ω

N , which is a polysymplectic form of rank N−1 (as can be seen, for


instance, by analyzing the criteria of Proposition 3.1), so that after repeating it N times,we arrive at the conclusion of the theorem. 2

We conclude this section with a digression on general vector-valued 2-forms: this willhelp us to appreciate the simplicity and usefulness of the concept of a polysymplectic formadopted in this thesis, in comparison with other definitions that preceded ours. Certainly,finding the right path within the jungle of possible notions was the most difficult part of thiswork.

Suppose that ω ∈∧2

V ∗ ⊗ T is an arbitrary T -valued 2-form on V . To begin with, werecall that according to the convention adopted in this work (see, for instance, equation (3.4)),the rank of an ordinary 2-form ω is equal to half the dimension of its support. Thus we have,for every t∗ ∈ T ∗,

rk(ωt∗

)= 1

2dim supp ωt∗ . (3.7)

Now note that the linear mapping

T ∗ −→∧2

V ∗

t∗ 7−→ ωt∗

(3.8)

induces, for every integer k > 1, a canonically defined linear mapping∨ kT ∗ −→

∧2kV ∗

P 7−→ P (ω)(3.9)

where we have identified the space∨ k

T ∗ of covariant symmetric tensors of degree k over Twith the space of homogeneous polynomials P of degree k on T . Explicitly, in terms of abasis ea | 1 6 a 6 n of T , with dual basis ea | 1 6 a 6 n of T ∗, we have

P = Pa1... akea1 ∨ . . . ∨ eak =⇒ P (ω) = Pa1... ak

ωa1 ∧ . . . ∧ ωak . (3.10)

This allows us to introduce the following terminology:

Definition 3.1 Let V and T be finite-dimensional vector spaces and let ω be a T -valued2-form on V . We say that ω has constant rank N if rk(ωt∗) = N for every t∗ ∈ T ∗ \ 0and that ω has uniform rank N if the linear mapping (3.10) is injective for k = N andidentically zero for k = N + 1.

Using multi-indices α = (α1, . . . , αn) ∈ Nn, we set

eα = (e1)α1 ∨ . . . ∨ (en)αn where (ea)αa = ea ∨ . . . ∨ ea (αa times)


and

ωα = (ω1)α1 ∧ . . . ∧ (ωn)αn where (ωa)αa = ωa ∧ . . . ∧ ωa (αa times)

to rewrite equation (3.10) in the form

P =∑|α|=k

Pα eα =⇒ P (ω) =∑|α|=k

Pα ωα . (3.11)

Since eα | |α| = k is a basis of∨ k

T ∗, requiring ω to have uniform rank N amounts toimposing the following conditions:

ωα | |α| = N is linearly independent

ωα = 0 for |α| = N + 1. (3.12)

It is in this form that the requirement of uniform rank appears in the definition of a poly-symplectic form adopted in Ref. [11].

To gain a better understanding for the conditions of constant rank and of uniform rankintroduced above, we note first of all that they both generalize the standard notion of rankfor ordinary forms. Indeed, when n = 1, that is, given an ordinary 2-form ω of rank N on V ,we can choose a canonical basis e1, . . . , eN , f1, . . . , fN of a subspace of V complementaryto the kernel of ω, with dual basis e1, . . . , eN , f1, . . . , fN of the subspace supp ω of V ∗,to conclude that ω = ei ∧fi and therefore

ωN = ± e1∧ . . . ∧ eN∧f1 ∧ . . . ∧fN 6= 0 , ωN+1 = 0 .

In other words, the rank of ω can be characterized as that positive integer N for whichωN 6= 0 but ωN+1 = 0. From this observation, it follows that, in the general case consideredbefore, the requirement of uniform rank implies that of constant rank because it guaranteesthat for every t∗ ∈ T ∗ \0, we have ωN

t∗6= 0 and ωN+1

t∗= 0 , that is, rk(ωt∗) = N . However,

the converse does not hold, as shown by the following

Example 3.1 ( n = 2 , N = 2 , dim V = 4 , ker ω = 0 ) :

Let V = R4 , T = R2 and consider the R2-valued 2-form ω built from the following twoordinary 2-forms:

ω1 = dx ∧ dy + du ∧ dv , ω2 = dx ∧ du − dy ∧ dv .

Then for t∗ = t∗a ea ∈ (R2)∗, we have

ωt∗ = t∗a ωa = dx ∧(t∗1 dy + t∗2 du

)+ dv ∧

(t∗2 dy − t∗1 du

).


Thus we obtain, for every t∗ 6= 0,

(ωt∗)2 ≡ ωt∗ ∧ ωt∗ =

((t∗1)

2 + (t∗2)2)

dx ∧ dy ∧ du ∧ dv 6= 0 ,

whereas, due to the fact that we are in a four-dimensional space,

(ωt∗)3 ≡ ωt∗ ∧ ωt∗ ∧ ωt∗ = 0 ,

which guarantees that ω has constant rank 2. However, ω does not have uniform rank 2,since

ω1∧ ω2 = 0 .

On the other hand, polysymplectic forms do have uniform rank:

Proposition 3.2 Let V and T be finite-dimensional vector spaces and let ω be a T -valuedpolysymplectic form of rank N on V . Then ω has uniform rank N .

Proof. Introducing a (polysymplectic) canonical basis in which ω assumes the form givenby equations (3.5) and (3.6), suppose that α = (α1, . . . , αn) ∈ Nn is a multi-index of degree k(i.e., such that |α| = α1 + . . . + αn = k) and consider the form

ωα = ±((ei11 ∧ . . . ∧ ei1α1 ) ∧ . . . ∧ (ein1 ∧ . . . ∧ einαn )

)∧(e1

i11∧ . . . ∧ e1

i1α1

)∧ . . . ∧

(en

in1∧ . . . ∧ en

inαn

).

Obviously any such form vanishes when k = N + 1 since it then contains an exteriorproduct of (N + 1) 1-forms ei belonging to an N -dimensional subspace. On the other hand,all these forms are linearly independent when k = N since ωα then contains the exteriorproduct e1 ∧ . . . ∧ eN multiplied by the exterior product of α1 1-forms of type e1

i with . . .

with αn 1-forms of type eni ; thus ωα and ωβ belong to different subspaces of

∧2NV ∗ whenever

α 6= β. 2

The converse statement, as we shall see soon, is remote from being true. In fact, if it weretrue, then if n > 2, it should be possible to construct the polylagrangian subspace as the sumof the kernels of the projected forms, as required by Proposition 3.1. Therefore, it shouldbe possible to show that the subspace defined as the sum of these kernels is isotropic. Andindeed, as a partial result in this direction, we have the following

Proposition 3.3 Let V and T be finite-dimensional vector spaces and let ω be a T -valued2-form of uniform rank N on V . Then for any t∗1, t

∗2 ∈ T ∗ \0 , the kernel of ωt∗1

is isotropicwith respect to ωt∗2

.


Proof. Given u, v ∈ ker ωt∗1, we have

iuωNt∗1

= N iuωt∗1∧ ωN−1

t∗1= 0 , ivω

Nt∗1

= N ivωt∗1∧ ωN−1

t∗1= 0 ,

and thereforeωt∗2

(u, v) ωNt∗1

= iuiv(ωt∗2

∧ ωNt∗1

)= 0 .

Using that ωNt∗16= 0, it follows that ω

t∗2(u, v) = 0. 2

However, isotropy of the subspace defined as the sum of the kernels of all the projected forms,which is equivalent to the (stronger) condition that for any t∗1, t

∗2, t

∗3 ∈ T ∗ \ 0 , ker ωt∗1

andker ωt∗2

are orthogonal under ωt∗3, i.e., that

ωt∗3(u1, u2) = 0 for u1 ∈ ker ωt∗1

and u2 ∈ ker ωt∗2,

cannot be derived from the condition of uniform rank. A nice counterexample is obtainedby choosing V and T to be the same space, supposing it to be a Lie algebra g and defining ωto be the commutator in g. Then for t∗ ∈ g∗, the kernel ker ωt∗ and the support supp ωt∗ ofthe projected form ωt∗ are the isotropy algebra of t∗ and the tangent space to the coadjointorbit passing through t∗, respectively. There is one and only one semisimple Lie algebra forwhich ω has constant rank, since this condition states that all coadjoint orbits except thetrivial one, 0, should have the same dimension: this is the algebra of type A1, that is, R3

equipped with the vector product ×.


Let V = T = R3 and consider the R3-valued 2-form ω built from the following three ordinary2-forms:

ω1 = dy ∧ dz , ω2 = dz ∧ dx , ω3 = dx ∧ dy .


ωt∗ = t∗a ωa = t∗1 dy ∧ dz + t∗2 dz ∧ dx + t∗3 dx ∧ dy .

Obviously, ω1, ω2 and ω3 are linearly independent and hence ω has uniform rank 1, sincethere exists no non-zero 4-form on a three-dimensional space. On the other hand, we have

ker ωt∗ = 〈 t∗1∂

∂x+ t∗2

∂

∂y+ t∗3

∂

∂z〉 ,

Therefore, the intersection of the three kernels ker ω1, ker ω2 and ker ω3 is 0 (i.e., ω isnon-degenerate). However, ker ω1 and ker ω2 are orthogonal under ω1 and under ω2 butnot under ω3. Now if there existed a polylagrangian subspace it would have to coincide withthe sum of the kernels of all the projected forms, but that is the whole space R3, which isnot isotropic. Thus ω does not admit a polylagrangian subspace.


Finally, we observe that even if the sum of the kernels of all the projected forms is an isotropicsubspace with respect to ω, it may still fail to be a polylagrangian subspace, as shown bythe following


Let V = R5, T = R2 and consider the R2-valued 2-form ω built from the following twoordinary 2-forms:

ω1 = dx1 ∧ dx4 + dx2 ∧ dx3 , ω2 = dx1 ∧ dx3 − dx2 ∧ dx5 .


ωt∗ = t∗a ωa = dx1 ∧(t∗1 dx4 + t∗2 dx3

)+ dx2 ∧

(t∗1 dx3 − t∗2 dx5

).

Obviously, ω1, ω2 and the forms

(ω1)2 ≡ ω1 ∧ ω1 = 2 dx1 ∧ dx2 ∧ dx3 ∧ dx4 ,

ω1 ∧ ω2 = dx1 ∧ dx2 ∧ dx4 ∧ dx5 ,

(ω2)2 ≡ ω2 ∧ ω2 = 2 dx1 ∧ dx2 ∧ dx3 ∧ dx5 ,

are linearly independent and hence ω has uniform rank 2, since there exists no non-zero6-form on a five-dimensional space. On the other hand, we have

ker ωt∗ = 〈 t∗1t∗2∂

∂x3− (t∗2)

2 ∂

∂x4+ (t∗1)

2 ∂

∂x5〉 .

The intersection of the two kernels ker ω1 and ker ω2 is 0 (i.e., ω is non-degenerate). Notethat their (direct) sum is the two-dimensional subspace of V , say L′, spanned by ∂/∂x4 and∂/∂x5, whereas the subspace spanned by all the kernels ker ωt∗ (t∗ ∈ T ∗ \ 0) is the three-dimensional subspace of V , say L′′, spanned by ∂/∂xi with i = 3, 4, 5, and this is isotropicwith respect to all the forms ωt∗ (t∗ ∈ T ∗ \ 0). More than that: since its codimension is 2,it is maximal isotropic with respect to all the forms ωt∗ (t∗ ∈ T ∗ \ 0). Now if there existeda polylagrangian subspace it would have to coincide with L′ and also with L′′, but these twoare not equal and do not have the right dimension, which according to equation (3.2) wouldhave to be 4: both of them are too small. Thus ω does not admit a polylagrangian subspace.

To summarize, the examples given above show that the hypothesis of existence of a poly-lagrangian subspace is highly non-trivial and very restrictive: as it seems, it cannot bereplaced by any other hypothesis that is not obviously equivalent. The examples also showthe great variety of possibilities for the “relative positions” of the kernels of the variousprojected forms that prevails when such a subspace does not exist. In this sense, the definition

3.2 Multisymplectic vector spaces 55

adopted in Ref. [11] is quite inconvenient, since it makes no reference to this subspace, thushiding the central aspect of the theory.

To conclude, we want to add some remarks about the relation between the polylagrangiansubspace, when it exists, and the more general class of maximal isotropic subspaces. First,we emphasize that in contrast with a polylagrangian subspace, maximal isotropic subspacesalways exist. To prove this, it suffices to start out from an arbitrary one-dimensional subspaceL1, which is automatically isotropic, and construct a chain L1 ⊂L2 ⊂ . . . of subspaces whereLp+1 is defined as the direct sum of Lp and the one-dimensional subspace spanned by somenon-zero vector in its 1-orthogonal complement Lω,1

p . For dimensional reasons, this processmust stop at some point, which means that at this point we have succeeded in constructing amaximal isotropic subspace. However, nothing guarantees that maximal isotropic subspacesresulting from different chains must have the same dimension, nor that there must existsome chain leading to a maximal isotropic subspace of sufficiently high dimension to bepolylagrangian: this happens only in the special case of ordinary forms (n = 1), whereall maximal isotropic subspaces have the same dimension and where the notions of a poly-lagrangian subspace (or simply lagrangian subspace, in this case) and of a maximal isotropicsubspace coincide.

Another important point concerns the relation between the notions of isotropic subspaceand maximal isotropic subspace with respect to the form ω and with respect to its projections.First, it is obvious that a subspace of V is isotropic with respect to ω if and only if it isisotropic with respect to each of the projected forms ω

t∗(t∗ ∈ T ∗ \ 0) or ωa (1 6 a 6 n).

However, this no longer holds when we substitute the word “isotropic” by the expression“maximal isotropic”: a subspace of V that is maximal isotropic with respect to each of theprojections of ω certainly will be maximal isotropic with respect to ω, but conversely, it canvery well be maximal isotropic with respect to ω (and hence isotropic with respect to each ofthe projections of ω) but even so fail to be maximal isotropic with respect to some of them.And finally, a polylagrangian subspace of V will certainly be maximal isotropic with respectto each of the projections of ω (this follows from Proposition 3.1), but as we have seen in thelast example above, the converse is false: a subspace can be maximal isotropic with respectto each of the projections of ω without being polylagrangian. These statements illustrate thespecial nature of the polylagrangian subspace, already in the case of vector-valued 2-forms.

3.2 Multisymplectic vector spaces

According to Definition 1.2, an (n+1)-form ω on a vector space W which is (n−1)-horizontalwith respect to a subspace V of W of codimension n in W is said to be multisymplectic ofrank N if it is non-degenerate and if there exists a multilagrangian subspace L of V ofcodimension N in V (and hence of codimension N + n in W ), characterized by the property


thatω[

V

(L)

=∧n

1 L⊥ ≡∧n

L⊥ ∩∧n

1W∗ . (3.13)

In this case, it is obvious that, according to equation (1.61) or (1.62),

dim L = Nn + 1 , (3.14)

and hencedim V = N(n + 1) + 1 , (3.15)

anddim W = (N + 1)(n + 1) . (3.16)

In particular, we conclude that the dimension of a vector space carrying a multisymplecticform has to be a multiple of its degree n + 1: this generalizes a well known property ofsymplectic forms, which are included as the special case n = 1 (2-forms).3 More generally,we have

rk(ω) = N =1

n + 1dim supp ω − 1 , (3.17)

which constitutes another example of the phenomenon already mentioned in the first chapter,according to which it may be convenient to use the term “rank” of a form for an expressionthat is numerically different from the dimension of its support but uniquely determined by it.

Given an (n + 1)-form ω on a vector space W which is (n− 1)-horizontal with respect toa subspace V of W of codimension n in W , we can define its symbol, which will be a 2-formon W taking values in the vector space T =

∧n−1T ∗, where T ∼= W/V . Note that the aux-

iliary space T has the same dimension as the base space T itself: n = n. Moreover, we knowthat if ω is multisymplectic, then ω will be polysymplectic, and the polylagrangian subspacefor ω will be the same as the multilagrangian subspace for ω. Comparing equations (3.14)and (3.2), we conclude that in this case

dim ker ω = 1 . (3.18)

In fact, since by hypothesis ω is non-degenerate, the linear mapping

ω[V : V −→

∧n1 W ∗

v 7−→ ivω, (3.19)

(see equation (1.53)) is injective and ker ω is exactly the pre-image, under ω[V , of the

subspace∧n

0 W ∗ ∼=∧n

T ∗ , which is one-dimensional.

When n > 2, these considerations, together with Proposition 3.1, lead to the followingcriterion to decide whether an (n + 1)-form is multisymplectic:

3Note that for n=1, the definition of a multisymplectic form reduces to that of a symplectic form, sincein this case, the condition of 0-horizontality is empty and that of existence of a lagrangian subspace L canbe derived as a theorem. The main difference between this case and those where n>1 is that here, L is notunique.

3.2 Multisymplectic vector spaces 57

Proposition 3.4 Let W be a finite-dimensional vector space and V be a fixed subspaceof W , with n = dim(W/V ) > 2, let ω be a non-degenerate (n− 1)-horizontal (n + 1)-formon W and let ω be the symbol of ω, which is a T -valued 2-form on V , where T ∼= W/V and

T =∧n−1

T ∗. Consider the subspace L of V spanned by the kernels of all the projections ωt

(t ∈ T ∗ \ 0) of ω:

L =∑

t∈ T ∗\0

ker ωt∗ .

For the form ω to be multisymplectic and L to be the pertinent multilagrangian subspace, thefollowing conditions are necessary and sufficient:

• L is isotropic;

• dim L = (n dim V + 1)/(n + 1).

To formulate the algebraic Darboux theorem in the specific context of multisymplecticforms, we begin by adapting Definition 1.6 to this context: explicitly, a (multisymplectic)Darboux basis or (multisymplectic) canonical basis is a basis

eµ, ei, ejν , e0 | 1 6 µ, ν 6 n , 1 6 i, j 6 N

of W , with dual basis

eµ, ei, eνj , e

0 | 1 6 µ, ν 6 n , 1 6 i, j 6 N

of W ∗, such that

• eµ | 1 6 µ 6 n is a basis of a subspace of W complementary to V , isomorphic to T ,and eµ | 1 6 µ 6 n is the corresponding dual basis of the subspace V ⊥ of W ∗, sothat if we define

e0 = e1 ∧ . . . ∧ en e eµ = ieµe0 (1 6 µ 6 n) ,

then e0 spans the one-dimensional subspace∧n

0 W ∗ ∼=∧n

T ∗ and eµ | 1 6 µ 6 n is a basis of the n-dimensional subspace

∧n−10 W ∗ ∼=

∧n−1T ∗ ;

• ei | 1 6 i 6 N is a basis of a subspace of V complementary to L ;

• eiµ, e0 | 1 6 µ 6 n , 1 6 i 6 N is a basis of L ;

• e0 spans the one-dimensional subspace ker ω ;


and such that

ω = ei ∧ eµi ∧ eµ + e0 ∧ e0 . (3.20)

and for the symbol

ω =(ei ∧ eµ

i

)⊗ eµ . (3.21)

Obviously, an (n + 1)-form ω ∈∧n+1

2 W ∗ that admits such a canonical basis is a multisym-plectic form. Conversely, as a corollary of Theorem 1.11, we have

Theorem 3.2 (Darboux theorem for multisymplectic vector spaces)

Every multisymplectic vector space admits a canonical basis.

Although this theorem has already been proved in the first chapter, we want to present analternative proof based on the analogous theorem for polysymplectic forms (Theorem 3.1),using the symbol. This procedure uses strongly the fact that the kernel of the symbol ofthe multisymplectic form is only one-dimensional, which allows to reduce the problem to thepolysymplectic case.

Proof. We begin by choosing an arbitrary basis eµ | 1 6 µ 6 n of a subspace of Wcomplementary to V , isomorphic to T , with dual basis eµ | 1 6 µ 6 n of the subspaceV ⊥ of W ∗, and define e0 = e1 ∧ . . . ∧ en and eµ = ieµ

e0, as above. We also define a 1-form

e0 on W by e0 = (−1)n ie1. . . ien

ω , that is,

〈e0, w〉 = ω(w, e1, . . . , en) for w ∈ W .

Since ω is non-degenerate, ker e0 ∩ ker ω = 0 . Now choose a vector e0 ∈ ker ω such that〈e0, e0〉 = 1 and pick a subspace W0 of W complementary to ker ω , setting V0 = V ∩W0

and L0 = L ∩W0 . Then the restriction of the symbol of ω of ω to V0 is a non-degenerate

polysymplectic form taking values in the auxiliary space T =∧n−1

0 W ∗ ∼=∧n−1

T ∗ ,with polylagrangian subspace L0, and so V0 admits a (polysymplectic) canonical basis ei, e

jν | 1 6 ν 6 n , 1 6 i, j 6 N for ω adapted to the basis eµ | 1 6 µ 6 n of T . Now

observing that the (n− 1)-horizontal (n + 1)-form

ω − e0 ∧ e0

is degenerate, with kernel equal to ker ω, but has the same symbol ω as the original (n−1)-horizontal (n + 1)-form ω, we conclude that eµ, ei, e

jν , e0 | 1 6 µ, ν 6 n , 1 6 i, j 6 N is

a (multisymplectic) canonical basis for ω. 2

3.3 Poly- and multisymplectic manifolds and fiber bundles 59

3.3 Poly- and multisymplectic manifolds

and fiber bundles

The Darboux thorems of the second chapter guarantee the existence of canonical local co-ordinates in which a polysymplectic form is expressed as

ω =(dqi ∧ dpa

i

)⊗ ea ,

and a multisymplectic form as

ω = dqi ∧ dpµi ∧ dnxµ − dp ∧ dnx ,

where dnxµ = i∂µdnx , thus reproducing precisely the fundamental canonical structure of

classical field theory.

Obviously, what has been achieved in this thesis is not the end but only the beginning.We hope that the theory of poly- and multisymplectic manifolds and fiber bundles, for whichwe believe to have laid the foundation, will in a not too distant future turn out to be asinteresting and fruitful as symplectic geometry.

Appendix A

Poincare Lemma

In this appendix, we state the version of the Poincare lemma used in the proof of the Darbouxtheorem at the end of Chapter 2.

Theorem A.1 Let ω ∈Ωk(P, T ) be a closed form on a manifold P taking values in a fixedvector space T and let L be an involutive distribution on P . Suppose that ω is (k − r)-horizontal (with respect to L), i.e., such that for any p ∈P and all v1, . . . , vr+1 ∈Lp, wehave

iv1 . . . ivr+1ωp = 0 .

Then for any point of P there exist an open neighborhood U of that point and a (k− 1)-formθ ∈Ωk−1(U, T ) on U which is also (k − r)-horizontal (with respect to L), i.e., such that forany p ∈U and all v1, . . . , vr ∈Lp, we have

iv1 . . . ivrθp = 0 ,

and such that ω = dθ on U .

Proof. Due to the local character of this theorem and since the subbundle L of TP isinvolutive, we can without loss of generality work in a local chart of the manifold P aroundthe chosen reference point in which the foliation defined by L is “straightened out”, so wemay assume that P ∼= K0⊕L0 with L ∼= L0 with fixed subspaces K0 and L0 and such thatthe aforementioned reference point corresponds to the origin. (In what follows, we shall omitthe index 0.) We also suppose that T = R, since we may prove the theorem separately foreach component of ω and θ, with respect to some fixed basis of T .

For t ∈R, define the “K-contraction” FKt : P → P and the “L-contraction” FL

t : P → Pby FK

t (x, y) = (tx, y) and FLt (x, y) = (x, ty) ; obviously, FK

t and FLt are diffeomorphisms if

61

62 Poincare Lemma

t 6= 0 and are projections if t = 0. Associated with each of these families of mappings thereis a time dependent vector field which generates it in the sense that, for t 6= 0,

XKt

(FK

t (x, y))

=d

dsFK

s (x, y)

∣∣∣∣s=t

and XLt

(FL

t (x, y))

=d

dsFL

s (x, y)

∣∣∣∣s=t

.

Explicitly, for t 6= 0,

XKt (x, y) = t−1(x, 0) and XL

t (x, y) = t−1(0, y) .

Define ω0 = (FL0 )∗ω and, for ε > 0,

θε =

∫ 1

ε

dt((FL

t )∗(iXLtω) + (FK

t )∗(iXKt

ω0))

,

as well as

θ = limε→0

θε =

∫ 1

0

dt((FL


t )∗(iXKt

ω0))

.

To see that the (k−1)-forms θ and θε are well defined, consider k−1 vectors (ui, vi) ∈K⊕L(1 6 i 6 k − 1) and observe that, for t 6= 0,

(FLt )∗(iXL

tω)(x,y)

((u1, v1), . . . , (uk−1, vk−1)

)= ω(x,ty)

((0, y), (u1, tv1), . . . , (uk−1, tvk−1)

),

and

(FKt )∗(iXK

tω0)(x,y)

((u1, v1), . . . , (uk−1, vk−1)

)= tk−1 ω(tx,0)

((x, 0), (u1, 0), . . . , (uk−1, 0)

).

Here we see easily that both expressions are differentiable in t and provide (k − 1)-formswhich are (k − r)-horizontal and (k − 1)-horizontal with respect to L, respectively. Thus,θε and θ are (k − 1)-forms which are (k − r)-horizontal with respect to L. Moreover, sincedω = 0 and dω0 = 0,

dθε =

∫ 1

ε

dt((FL


t )∗(iXKt

ω0))

=

∫ 1

ε

dt((FL

t )∗(LXLtω) + (FK

t )∗(LXKt

ω0))

=

∫ 1

ε

dt( d

dt

((FL

t )∗ω)

+d

dt

((FK

t )∗ω0

))= ω − (FL

ε )∗ω + ω0 − (FKε )∗ω0 .

Taking the limit ε→ 0, we get (FLε )∗ω → ω0 and (FK

ε )∗ω0 → 0 and hence dθ = ω. 2

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