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SYMPLECTIC GEOMETRY AND HILBERT’S FOURTHPROBLEM

J.C. ALVAREZ PAIVA

Abstract. Inspired by Hofer’s definition of a metric on the space ofcompactly supported Hamiltonian maps on a symplectic manifold, thispaper exhibits an area-length duality between a class of metric spacesand a class of symplectic manifolds. Using this duality, it is shown thatthere is a twistor-like correspondence between Finsler metrics on RP n

whose geodesics are projective lines and a class of symplectic forms onthe Grassmannian of 2-planes in Rn+1.

... es quiza un error suponer que puedan inventarse

metaforas. Las verdaderas, las que formulan ıntimas

conexiones entre una imagen y otra, han existido siem-

pre ... .Jorge Luis Borges.

Contents

1. Introduction 12. Finsler manifolds and spaces of geodesics 43. From symplectic forms to distance functions 94. Reduction to two dimensions 185. A characterization of admissible symplectic forms 21References 24

1. Introduction

Hilbert’s fourth problem asks to construct and study all metrics on Rn

such that the straight line segment is the shortest curve joining two points.Busemann, who called these metrics projective, proposed an integral-geometricconstruction which is inspiringly simple.

Let us say that a (possibly) signed measure on the set of hyperplanesis quasi-positive if whenever x, y, and z are three non-collinear points, themeasure of the set of hyperplanes intersecting twice the wedge formed by thesegments xy and yz is strictly positive. Given a quasi-positive measure suchthat the set of all hyperplanes passing through a point has measure zero,

1991 Mathematics Subject Classification. 53D25; 53C65, 53C60, 53C70.Key words and phrases. Symplectic geometry, Hofer metric, integral geometry,

Hilbert’s fourth problem, Finsler metric, G-spaces.1

2 J.C. ALVAREZ PAIVA

we define the distance between two points as the measure of all hyperplanesintersecting the line segment joining them.

It is easy to see that the metric defined by Busemann’s construction isprojective. It is more difficult to see that every projective metric that issufficiently smooth is of this form. This was proved by Pogorelov in [27] (seealso [29] and [9] for alternate presentations).

While it is possible to take Busemann’s construction and Pogorelov’s the-orem as the solution (and the end) of Hilbert’s fourth problem, there arestrong reasons for not doing so. For one, a large number of projective metricscannot be constructed by taking positive measures on the space of hyper-planes and it is not clear how to construct all quasi-positive measures. Asecond objection is that Busemann’s construction, by itself, does not shedmuch light on the properties of projective metrics. Therefore, it is useful tocomplement the results of Busemann and Pogorelov with other characteri-zations of projective metrics that either yield new explicit examples, or shedmore light on the general properties of these metric spaces.

The main result of this paper is a new characterization of projective met-rics in terms of a class of symplectic forms on the space of lines:

Following [19], we say that a differential two-form ω on the space of ori-ented lines in Rn is admissible if

(1) it is closed;(2) the pull-back of ω to the submanifold of all oriented lines passing

through an arbitrary point in Rn is identically zero;(3) it is odd (i.e., if a is the involution than reverses the orientation of

the lines, then a∗ω = −ω).

Let ω be a differential two-form on the space of oriented lines in Rn. If xand y are two points in Rn, and Π is a plane containing them, set dω(x, y; Π)to be the integral of |ω| over the set of all oriented lines on Π intersectingthe segment xy.

Theorem 1.1. If ω is an admissible symplectic form, then the functiondω is independent of the choice of 2-dimensional subspace and defines aprojective Finsler metric on Rn. Moreover, any projective Finsler metriccan be obtained from this construction.

The usefulness of this new characterization is threefold: together with adetailed analysis of Crofton-type formulas, it allows us to extend the classicalCrofton formulas to all projective Finsler spaces ([7]); it uncovers the relationbetween Hilbert’s fourth problem and the “black and white” twistor theoryof Guillemin and Sternberg ([20, 3]); and it opens the way for symplecticand contact techniques a l’Arnold in the study of submanifolds in projectiveFinsler spaces ([4]).

Theorem 1.1, which was announced in [9], is a relatively easy consequenceof the following symplectic result.

SYMPLECTIC GEOMETRY AND HILBERT’S FOURTH PROBLEM 3

Theorem 1.2. Let ω be an admissible two-form on the space of orientedlines in Rn, n > 2. The form ω is symplectic if and only if for everyplane Π ⊂ Rn the pull-back of ω to the (two-dimensional) submanifold of alloriented lines lying on Π never vanishes.

When n = 3, the above result has a simple topological proof that onlymakes use of the non-degeneracy of ω (see [3]), however this proof completelybreaks down for n > 3. The present proof of the general case of theorem 1.2relies on a certain duality between metric and symplectic geometry whichextends Arnold’s area-length duality for spherical curves (see [11]). Curiouslyenough, the area-length duality by itself almost provides an independentproof of theorem 1.1. The trouble is that, a priori, the metrics obtainedthrough the construction in section 3 are not Finsler metrics, but just G-space metrics in the sense of Busemann ([18]). On the other hand, theconstruction is a very general criterion for the metrization of path geometriesthat is interesting in its own right.

Let M be an n-dimensional manifold together a prescribed system ofsmooth curves such that through every point and every direction there isa unique curve passing through this point in the given direction. Such ageometric structure is called a path geometry on M . We shall say that apath geometry is tame if it satisfies the following two properties:

(1) any two distinct points in M that are sufficiently close are joined bya unique curve in the system;

(2) the system of oriented curves on M obtained from the path geom-etry by providing each curve with its two possible orientations isparameterized by a smooth 2n− 2 dimensional manifold Γ.

Note that whenever we have a tame path geometry the manifold M pa-rameterizes a system of submanifolds in the parameter space Γ: to eachx ∈ M we associate the submanifold x of all curves passing through x.When the path geometry is the system of geodesics of a Finsler metric onM , it is known (see [2], [14] and section 2) that the space of geodesics, Γ,carries a natural symplectic form and that the x, x ∈ M , are Lagrangiansubmanifolds. In section 3 the reader will find the following partial converse:

Theorem 1.3. Let M be smooth manifold of dimension greater than twoendowed with a tame path geometry and let Γ denote its parameter space. Ifthere exists a symplectic form on Γ such that the submanifold of all curvespassing through an arbitrary point in M is Lagrangian, then M can be given aG-space metric such the path geometry coincides with the system of geodesicsin M .

Roughly, the idea of the proof is to consider the map x 7→ x as an em-bedding of M into the space of all Lagrangian submanifolds of Γ and usethe Hofer metric in this infinite-dimensional space to induce a metric inM . More precisely, infinitesimal Hamiltonian deformations of a Lagrangian


submanifold L in the symplectic manifold (Γ, ω) are described as the quo-tient of all smooth functions on L by those that are constant. This space,C∞(L)/constants, carries the infinitesimal Hofer norm

‖f‖ = variation(f) = max f −min f.

Since the family of Lagrangian spheres x, x ∈ M , induces canonical linearmaps

TxM −→ C∞(x)/constants,we can pull-back the Hofer norm to TxM . This gives us, a priori, a low-regularity version of a Finsler structure on M . Actually, in order to avoidsome technical difficulties (e.g., it is not known when a low-regularity Finslermetric on a manifold M gives rise to a G-space structure) and to makethe proof more accessible to metric and Finsler geometers, we present anintegrated version of the above construction.

The breakdown of the paper — and the global structure of the proof oftheorem 1.1 — is as follows: section 2 reviews some basic facts on Finslermanifolds and their spaces of geodesics, and ends with a proof of the (easy)second half of theorem 1.1. Section 3 contains the proof of theorem 1.3 andof the following weaker version of theorem 1.2:

Theorem 1.4. If ω is an admissible symplectic form on the space of orientedlines in Rn, n > 2, then for every plane Π ⊂ Rn the pull-back of ω to the(two-dimensional) submanifold of all oriented lines lying on Π never changessign, nor vanishes on an open subset.

In section 4, the Busemann-Pogorelov solution of Hilbert’s fourth problemin dimension two is used to show that the first half of theorem 1.1 followsfrom theorem 1.2. In section 5, theorem 1.4 and a detailed analysis of thegeneralized conformal structure on the Grassmannian of oriented two-planesin Rn+1, n > 2, are used to prove theorem 1.2, thus finishing the proof ofour main result.

Acknowledgments . The author warmly thanks I.M. Gelfand for his partic-ipation in the early stages of this work and gratefully acknowledges helpfulinteractions with G. Berck, E. Fernandes, and R. Schneider. He also thanksthe anonymous referee for the short sketch of the proof of theorem 1.3 givenin this introduction.

2. Finsler manifolds and spaces of geodesics

A Finsler manifold is a manifold together with the choice of a norm oneach tangent space. The precise definition requires us to restrict the class ofnorms to those where the unit sphere is smooth and quadratically convex :the principal curvatures are positive for some (and therefore any) auxiliaryEuclidean structure. These norms are intrinsically defined as follows:

Definition 2.1. A norm ϕ : V → [0,∞) is said to be a Minkowski norm ifthe unit sphere in V and its polar in V ∗ are smooth. A finite-dimensional


vector space provided with a Minkowski norm will be called a Minkowskispace.

Definition 2.2. Let M be a smooth manifold and let TM \ 0 denote itstangent bundle with the zero section deleted. A Finsler metric on M is asmooth function

ϕ : TM \ 0 −→ Rsuch that for each point m ∈ M the restriction of ϕ to TmM is a Minkowskinorm.

Given a Finsler metric ϕ on a manifold M , the length of a smooth curveγ : [a, b] → M is defined by the equation

length of γ :=∫ b

aϕ(γ(t))dt ,

and the distance between two points x and y in M is defined as the infimumof the lengths of all smooth curves joining x and y.

Later in the paper we shall be studying projective Finsler metrics: Finslermetrics defined on open convex subsets of RPn for which the geodesics lieon projective lines. Among the earliest examples of projective Finsler spacesare the Hilbert geometries:Hilbert geometries. Let D ⊂ Rn be a bounded, convex, open subset. Let xand y denote two distinct points on D and let a and b denote the two pointsof intersection of the boundary of D with the line passing through x and y(Fig. 1).

a

x

y

b

Figure 1.

The distance between x and y is defined by the formula:

d(x, y) :=12

ln(‖y − a‖‖x− a‖

‖x− b‖‖y − b‖

).

When the boundary of D is smooth and quadratically convex, the metricon D comes from a Finsler metric ϕ : TD \ 0 −→ R. Let v be a nonzerotangent vector at a point x ∈ D and let t1 and t2 be two positive realnumbers such that x + t1v and x − t2v belong to the boundary of D. Thevalue of ϕ at v is 1

2(t−11 + t−1

2 ).

Finsler geometry from the Hamiltonian viewpointLet M be a Finsler manifold and let ‖ · ‖x denote the norm at a tangent

space TxM . The Hamiltonian of a Finsler metric is the function H : T ∗M \0 → R whose value at a covector px is ‖px‖∗x, where ‖ · ‖∗x is the norm dual


to ‖ · ‖x. The set {p ∈ T ∗M : H(p) < 1} and its boundary are respectivelycalled the unit co-disc bundle and the unit co-sphere bundle of M and aredenoted by D∗M and S∗M .

Definition 2.3. If we denote by α the pullback of the canonical 1-form toS∗M , we define the Reeb vector field , X, by the equations

dα(X, ·) = 0, α(X) = 1.

The image of an integral curve of X under the Legendre transform —the map that sends a unit covector px ∈ T ∗xM to the unique unit vectorvx ∈ TxM such that px(vx) = 1 — is the curve of unit tangent vectorsof a geodesic parameterized with unit speed. Moreover, the length of anysegment of this geodesic is given as the integral of α over the correspondingsection of the integral curve of the Reeb vector field.

Spaces of geodesicsIf (M,ϕ) is a Finsler manifold such that its space of oriented geodesics is

a manifold G(M), then this manifold carries a natural symplectic structure.Indeed, let S∗M denote the unit co-sphere bundle of M and let π : S∗M →G(M) be the canonical projection which sends a given unit covector to thegeodesic which has this covector as initial condition. If i : S∗M → T ∗Mdenotes the canonical inclusion into T ∗M and ω0 is the standard symplecticform on T ∗M , there is a unique symplectic form ω on G(M) that satisfiesthe equation π∗ω = i∗ω0.

Notice that the space of oriented projective lines in RPn is the Grassman-nian of oriented 2-planes in Rn+1, G+

2 (Rn+1), while the space of orientedlines in Rn is easily identifiable with the tangent (or cotangent) bundle ofSn−1.

The Huygens correspondenceLet M be a Finsler manifold with a smooth manifold of geodesics G(M).

The Huygens correspondence is the double fibration

S∗Mπ1

||yyyy

yyyy

yπ2

$$IIIIIIIII

M G(M),

where π1 and π2 are the canonical projections. This correspondence allowsus to relate the metric geometry of M , the symplectic geometry of G(M),and the contact geometry of S∗M . We will now review some simple aspectsof the geometry of the fibrations

π1 : S∗M → M and π2 : S∗M → G(M).

Definition 2.4. Let M be an n-dimensional Finsler manifold with unit co-sphere bundle S∗M and canonical form α. The contact structure on S∗Mis the field of hyperplanes in TS∗M defined by the equation α = 0. An


immersed manifold L ⊂ S∗M is said to be Legendrian if its dimension isn− 1 and the form α vanishes on every one of its tangent spaces.

For example, the fibers of the projection π1 are Legendrian. In the ter-minology of Arnold (see [13]) the fibration

π1 : S∗M −→ M

is a Legendrian fibration.We now turn our attention to the fibration π2 : S∗M → G(M). Its

most important property is that it is a line or circle bundle over G(M)with connection form α and curvature dα = π∗2ω. In other words, it is aprequantization of the symplectic manifold (G(M), ω). Other simple anduseful properties are given in the following propositions:

Proposition 2.1. The map a : G(M) → G(M) that sends each oriented ge-odesic to the same geodesic with its opposite orientation is an anti-symplecticinvolution.

Proof. Note that the involution A that sends every unit covector px to itsopposite −px takes the form α to −α and that π2 ◦ A = a ◦ π2. It followsthat

π∗2a∗ω = A∗π∗2ω = A∗dα = −dα.

This implies that π∗2(−a∗ω) = dα, which in turn implies that −a∗ω = ω. ¤

Proposition 2.2. If i : L → S∗M is a Legendrian immersion, then thecomposite map π2 ◦ i : L → G(M) is a Lagrangian immersion. In particular,the submanifold of all geodesics passing through a point in M is Lagrangian.

Proof. Note that the kernel of the differential of the projection π2 is transver-sal to the contact hyperplanes given by the equation α = 0. Since thetangent spaces of any Legendrian submanifold are contained in the contacthyperplanes, it follows that if i : L → S∗M is a Legendrian immersion,then π2 ◦ i is an immersion. To see that it is a Lagrangian immersion justcompute:

(π2 ◦ i)∗ω = i∗π∗2ω = i∗ωM = di∗α = 0.

¤

Proposition 2.3. If N ⊂ M is a totally geodesic submanifold, then the setof all geodesics lying on N is a symplectic submanifold of G(M).

Proof. Let i : N → M be the inclusion map. For each point x ∈ N the dualof the differential map

i∗x : TxN −→ TxM

is a surjective linear map i∗x : T ∗xM → T ∗xN . The restriction of this map tothe unit co-sphere S∗xM has a fold-type singularity along a submanifold Σx.Since N is isometrically embedded, i∗x defines a diffeomorphism between Σx

and the unit co-sphere S∗xN .


Performing this construction at each point of N we construct an embed-ding

j : S∗N −→ S∗M,

which can be characterized by saying that if px is a unit covector on N basedat x, then j(px) is the unique unit covector on M based at x for which

j(px) · i∗x(vx) = px · vx, for all vx ∈ TxN.

If αN and αM denote, respectively, the canonical one-forms on S∗N andS∗M , it is easy to check that j∗αM = αN . Moreover, when N is totallygeodesic, the map j is equivariant with respect to the actions defined by thegeodesic flows. As a result, we have the following commutative diagram:

S∗Nj //

πN

²²

S∗M

πM

²²G(N)

j // G(M),

where the map j : G(N) → G(M) is the natural embedding (i.e., a geodesicon N is a geodesic on M). This embedding is symplectic. Indeed, if ωN andωM are, respectively, the symplectic forms on G(N) and G(M), we havethat j∗ωM = ωN . To see this, note that

π∗N (j∗ωM ) = j∗(π∗MωM ) = j∗(dαM ) = dαN .

Since ωN is defined uniquely by the equality π∗NωN = dαN , we must havethat j∗ωM = ωN . ¤

Applying the previous results we obtain that a projective Finsler metricon RPn (or, at some cost of notation, on an arbitrary open convex subset ofRPn) induces a symplectic form ω on the space of oriented lines, G+

2 (Rn+1),with the following properties:

(1) the form ω is admissible,(2) the submanifold of all oriented lines lying on an arbitrary projective

plane is symplectic.We shall show in section 4 that there is a natural bijection between the

space of symplectic form on G+2 (Rn+1) satisfying these two properties and

the space of projective Finsler metrics on RPn. In section 5 we will seethat property (2) actually follows from property (1)and the fact that ω issymplectic.

For now let us prove the second, easier half, of theorem 1.1: consider aprojective Finsler metric on Rn and let ω be the associated symplectic two-form on the space of oriented lines in Rn. If x and y are two points in Rn,and Π is a plane containing them, set dω(x, y; Π) to be the integral of |ω|over the set of all oriented lines on Π intersecting the segment xy.

Proposition 2.4. Given two points x and y in Rn and any plane Π ⊂Rn containing them, the quantity dω(x, y; Π) equals four times the Finsler


distance between x and y. In particular, dω is independent of the choice ofplane.

Proof. Notice that since the metric is projective, the plane Π is totally ge-odesic and the distance between x and y in Π equals the distance betweenthem in Rn. Moreover, by proposition 2.3, the natural symplectic form onG(Π) is the pull-back of ω to the submanifold of all oriented lines lying on Π.We can now use the Crofton formula for Finsler surfaces due to Blaschke (see[15] and [2]) to write the distance between x and y as one-fourth the integralof |ω| over the set of all oriented lines intersecting the segment xy. ¤

3. From symplectic forms to distance functions

In this section we study a well-known open inverse problem in varia-tional calculus: given a system of smooth curves on a manifold M suchthat through every point and every direction there is a unique curve pass-ing through this point in the given direction, find necessary and sufficientconditions for this system to be the geodesics of some Finsler metric on M .

In order to describe the main theorem in this section, we first give for-mal definitions of the systems of curves and of the generalization of Finslermetrics that we will be considering.

Path geometries

Definition 3.1. Let M be a smooth manifold and let π : PTM → M be itsprojectivized tangent bundle. A path geometry on M is a smooth foliationof PTM by one-dimensional submanifolds that are transverse to the fibersPTxM , x ∈ M .

The projection onto M of a leaf of the path geometry will be called a path.Notice that the paths form a system of smooth curves such that throughevery point and every direction there is a unique curve passing through thispoint in the given direction. In what follows we will also make use of orientedpaths and path segments.

Definition 3.2. We shall say that a path geometry on a manifold M istame if it satisfies the following two properties:

(1) any two distinct points in M that are sufficiently close determine aunique path;

(2) the set of oriented paths on M , obtained from the path geometry byproviding each path with its two possible orientations, is parameter-ized by a smooth 2n− 2 dimensional manifold Γ.

Examples of tame path geometries are the geodesic foliations of rank-onesymmetric spaces and Hadamard manifolds (i.e., simply connected Riemann-ian manifolds of non-positive sectional curvature). However, most geodesicfoliations are not tame. For example, it will be shown later that if a pathhas a self-intersection, the path geometry cannot be tame.


Whenever we have a tame path geometry, the incidence relation

{(x, γ) ∈ M × Γ : x ∈ γ}is naturally diffeomorphic to the spherized (or homogeneous) tangent bundleof M , STM , and we have the double fibration

STMπ1

{{wwwwwwwwπ2

##FFFF

FFFF

F

M Γ.

Busemann’s G-spaces

In his researches on geodesics and inverse problems in variational calculus,Busemann (see [18]) abstracted the properties of Finsler spaces and defineda more general class of metric spaces which he called G-spaces or geodesicspaces.

Definition 3.3. A metric space (M,d) is said to be a G-space if it satisfiesthe following properties:

(1) The space M is locally compact.(2) Given two distinct points x and z, there exists a third distinct point

y for which d(x, y) + d(y, z) = d(x, z).(3) For every point p ∈ M there is a metric ball B centered at p such

that if x and y are in B, then there exists a point z such thatd(x, y) + d(y, z) = d(x, z).

(4) If x, y, z1, and z2 are points such that

d(x, y) + d(y, z1) = d(x, z1) = d(x, y) + d(y, z2) = d(x, z2),

then z1 = z2.

Property (2) states that the metric is intrinsic: the infimum of the lengthsof all curves joining a pair of points equals the distance between them.Property (3) implies that geodesic segments can be locally extended, whileproperty (4) implies that the extension of a geodesic segment is unique.

Roughly speaking, the main result in this section is a sufficient conditionfor a tame path geometry to be the “geodesic foliation” of a G-space.

Metrizations of tame path geometries

Theorem 3.1. Let M be smooth manifold of dimension greater than twoendowed with a tame path geometry and let Γ denote its parameter space. Ifthere exists a symplectic form on Γ such that the submanifold of all curvespassing through an arbitrary point in M is Lagrangian, then the system ofpaths in M is the system of geodesics for a G-space metric on M .

Later in this section it will be shown that if M is a Finsler manifold withmanifold of geodesics G(M) = Γ and ω is the induced symplectic form onG(M), then the constructed metric is the Finsler metric on M .


The motivation for the proof of theorem 3.1 in terms of the Hofer met-ric on the space of Lagrangian submanifolds of symplectic manifold hasalready been given in the introduction. However, the proof below is com-pletely elementary and only makes use of the following two classical resultsof Whitehead and Weinstein:

Definition 3.4. Let M be a manifold provided with a path geometry. Anopen set U ⊂ M is said to be convex if for any two points in U there is aunique path segment joining them and lying entirely inside U .

Theorem 3.2 (Whitehead, [31]). If M is a manifold provided with a pathgeometry, then every point in M has a convex neighborhood.

Theorem 3.3 (Weinstein, [30]). If L is an embedded Lagrangian manifoldof a symplectic manifold (Γ, ω), then there is a symplectomorphism betweena tubular neighborhood of L and a tubular neighborhood of the zero sectionin T ∗L that takes L to the zero section.

Using Whitehead’s theorem we can see that tame path geometries arereally quite special.

Proposition 3.1. If Γ is the parameter space of a tame path geometry onM , then

(1) all paths are simple,(2) for every point x ∈ M the submanifold of all oriented paths passing

through x, x, is an embedded sphere in Γ, and(3) whenever two points x and y are sufficiently close, then x and y

intersect in precisely two points.

Proof. Suppose a path γ has a double point x. By taking a sufficiently smallconvex neighborhood, V , around x, we see that there are pairs of points onγ that are joined by two or more paths (Fig. 2).

x

V

γ

Figure 2.

To prove the second claim, we first notice that x is immersed. Indeed, ifπ1 and π2 are the canonical projections of {(x, γ) ∈ M × Γ : x ∈ γ} onto Mand Γ, then x is the projection of π−1

1 (x) onto Γ. Since the kernels of thedifferentials of π1 and π2 are transversal, it follows that x is immersed. Thefirst claim implies that this immersion is injective.

The third claim is a restatement of the hypothesis. ¤


For the rest of the section we assume that M and Γ satisfy the hypothesisof theorem 3.1. The first step in the construction of the G-space metric onM is to define an open covering of M and an intrinsic metric on each opensubset, Vx, of the covering such that the geodesics for this metric coincidewith the path segments lying in Vx.

Lemma 3.1. There exists an open covering of M that associates to everypoint x ∈ M an open neighborhood Vx with the following properties:

(1) The set Vx is convex.(2) Any two distinct points in Vx determine a unique path.(3) There exists a symplectomorphism Ψx between a tubular neighbor-

hood U of x in Γ and a tubular neighborhood of the zero section inT ∗x that takes x to the zero section and such that if y belongs tothe closure of Vx, then y ⊂ U and Ψx(y) ⊂ T ∗x is the graph of theexterior differential of some smooth function fx

y on x.

Proof. From theorem 3.2 and the tameness of the path geometry, it is clearthat around every point x we can find an open set Vx satisfying the firsttwo properties. To see that we can choose Vx to satisfy the third propertyas well, we use theorem 3.3 and notice that if a point y is sufficiently closeto x, the Lagrangian sphere y is C1 close to x. Therefore, Ψx(y) ⊂ T ∗x isthe graph of a closed 1-form. Since x is a sphere of dimension greater thanone, it is simply connected and the closed 1-form is the exterior differentialof some smooth function. ¤

On each Vx we define a metric by setting

δx(y, z) :=12

[max(fx

y − fxz )−min(fx

y − fxz )

].

In other words, we consider the map y 7→ fxy as a mapping from Vx to

the space of smooth functions on x modulo constants and pull-back thedistance function ∆(f, g) := [max(f − g)−min(f − g)] /2. The metric ∆comes from the norm ‖f‖ = [max(f)−min(f)] /2 that symplectic geometerswill recognize as the local model for the Hofer norm. Like in many othernormed spaces, the geodesic segment joining two given points is far frombeing unique. However, it is easy to verify if three points belong to a geodesicsegment:

Lemma 3.2. Let f and g be two continuous functions on a compact topo-logical space X. The equality

max(f + g)−min(f + g) = max(f)−min(f) + max(g)−min(g)

holds if and only if there exist points x and y in X for which max(f) = f(x),max(g) = g(x), min(f) = f(y), and min(g) = g(y).

Proposition 3.2. The metric space (Vx, δx) is a length space where thegeodesics coincide with the prescribed system of curves on Vx ⊂ M .


Proof. A simple way of verifying whether a locally compact metric space isa length space is to see whether for any two point u and v there exists athird point w such that the distance between u and v equals the distancebetween u and w plus the distance between w and v (see [18], pp. 29).

In our case, let u and v be two points in Vx and let w be some point inthe unique curve segment joining u and v. Because the spheres u, v, and wintersect at the same two points, the functions fx

v −fxu , fx

v −fxw, and fx

w−fxu

have the same two critical points a and b in x. It is important to note thatunless w coincides with u or v none of these functions is constant.

Without loss of generality, assume that max(fxv − fx

u ) = fxv (a) − fx

u (a).By continuity, and because there are only two critical points, if w is closeto u we have that max(fx

v − fxw) = fx

v (a) − fxw(a). Moreover, as we move

w from u to v along the curve this cannot change since fxv − fx

w is neverconstant. The same argument applies to fx

w − fxu and we have that for any

w between u and v

max(fxv − fx

u ) = max(fxv − fx

w) + max(fxw − fx

u ).

Since we have the analogous statement for the minima, we conclude thatδx(u, v) = δx(u,w) + δx(w, v).

This argument not only proves that (Vx, δx) is a length space, but alsothat the path segment joining u and v is a geodesic segment for the metric.To complete the proof, we must show that any geodesic segment lies on apath. In order to do this, we prove that if w is not in the curve segmentjoining u and v, then δx(u, v) < δx(u,w) + δx(w, v).

We distinguish two cases: either w is not on the path passing throughu and v or w lies on this path, but does not belong to the segment thatjoins u and v. In the first case, the Lagrangian spheres u, v, and w donot intersect in the same two points, and by lemma 3.2 we have the stricttriangle inequality. In the second case, the Lagrangian spheres u, v, and wdo intersect in the same two points, but it is easily ascertained that whilefx

v − fxu reaches its maximum at a ∈ x, either fx

w− fxu or fx

v − fxw reaches its

maximum at b. Again, lemma 3.2 implies that we have the strict triangleinequality. ¤

This finishes the first step of the proof of theorem 3.1. The second stepconsists in piecing together the metrics we just constructed on the elementsof the open cover of M to obtain, first, a length structure and, then, aG-space metric on M .

Proposition 3.3. If x and y are two points on M for which Vx ∩ Vy is notempty, then δx and δy agree on the intersection.

Proof. The proof follows immediately from the following symplectic inter-pretation of the distance functions: if u and v are in Vx∩Vy, the Lagrangianspheres u and v are in tubular neighborhoods of x and y that are sym-plectomorphic to neighborhoods of the zero section of the cotangent of the


(n − 1)-dimensional sphere. Since n > 2, these tubular neighborhoods aresimply connected.

Take a simple curve on u joining the two points of intersection of u and vand concatenate it with a simple curve on v joining the same two points toobtain a loop σ. The distances δx(u, v) and δy(u, v) are both equal to one-half the absolute value of the symplectic area of any disc whose boundary isσ. The choice of σ is also irrelevant since the spheres u and v are Lagrangian.

¤

As a result of this proposition we have an open covering of M togetherwith a length structure — a way of measuring the length of curves — `x

defined on each set Vx of the covering. The length structures coincide onthe intersections and thus there is a unique length structure ` on M whichrestricts to `x on Vx for all x ∈ M . We define a pseudo-metric d on M bysetting

d(x, y) := inf{`(σ) : σ is a smooth curve joining x and y}Proposition 3.4. The pseudo-metric d defined above is actually a metric.Moreover, its geodesics coincide with the prescribed system of curves on M .

Proof. First let us prove that d is a metric. Let x and y be two distinctpoints on M and let V ′

x ⊂ Vx be a neighborhood of x whose closure iscompact and does not contain y. Since δx is continuous on Vx × Vx and theboundary of V ′

x is compact, we have that the distance between it and x isa positive number ε. Since any continuous curve between x and y must cutthe boundary of V ′

x, the length of this curve must be at least equal to ε > 0and hence the distance between x and y is strictly positive.

Now we show that if two points u and v are sufficiently close to a pointx, then d(u, v) = δx(u, v). In view of proposition 3.2 and the locality of thedefinition of geodesics, this implies that the geodesics of (M, d) are preciselythe paths on M .

Let V ′x ⊂ Vx be a neighborhood of x such that, for the metric δx, its

diameter is strictly less than twice the distance between its boundary andthat of Vx. If u and v are in V ′

x then the length of any continuous curvejoining these two points and leaving Vx is greater that δx(u, v). Since (Vx, δx)is a path metric space, we have that the infimum of the lengths of all curvesjoining u and v is δx(u, v) and hence d(u, v) = δx(u, v). ¤

To finish the proof of theorem 3.1, we must show that (M, d) is a G-space.This follows at once from the fact that (M,d) is a locally compact lengthspace whose geodesics agree with the paths of a path geometry on M .

Application to Finsler geometry

We now show that when the path geometry is the system of geodesics ofa Finsler metric on M and ω is the induced symplectic form on the spaceof geodesics G(M) = Γ, then the metric d defined above is the Finsler


metric on M . This is a generalization of Blaschke’s Crofton formula fortwo-dimensional Finsler spaces (see [15] and [2]).

Theorem 3.4. Let M be a Finsler manifold whose space of geodesics is asmooth manifold G(M). If every two points of M that are sufficiently closedetermine exactly one geodesic and if ω is the natural symplectic form onG(M), then the metric d constructed from ω is the original Finsler metricon M .

Proof. It is sufficient to show that on each Vx the Finsler metric agrees withδx. This implies that the length structure agrees with that of the Finslermetric and, therefore, the metrics agree.

Recall the symplectic interpretation of δx explained in the proof of propo-sition 3.3. If σ is a loop made up of two simple curves on u and v joiningtheir points of intersection and D is any disc with boundary equal to σ, thenδx(u, v) is the absolute value of the integral of ω over D. We can lift σ toa loop σ′ on the unit co-sphere bundle of Vx by concatenating the followingcurves (Fig. 3):

(1) the Legendre transforms of the unit vectors of all geodesics passingthrough u and belonging to σ,

(2) the Legendre transforms of the unit vectors tangent to the uniqueoriented geodesic going from u and v,

(3) the Legendre transforms of the unit vectors of all geodesics passingthrough v and belonging to σ, and

(4) the Legendre transforms of the unit vectors tangent to the uniqueoriented geodesic going from v and u.

u v

Figure 3.

If D′ is a disc on the unit co-sphere bundle of Vx spanning σ′ and π is thestandard projection onto the space of geodesics, then

2δx(u, v) =∫

π(D′)ω =

∫

D′π∗ω =

∫

D′dα =

∫

σ′α.

Notice that the integral of α over the curves 1 and 3 is zero, while its integralover the curves 2 and 4 equals the length of the geodesic segment. We havethen that δx(u, v) is the Finsler distance between u and v. ¤

Positivity properties of admissible two-forms

To finish the section we apply theorem 3.1 to the case where M = Rn andΓ = G(Rn) is the space of oriented lines in Rn. In this case, the precedingconstructions give us a way to construct projective metrics (i.e., G-spacemetrics on Rn for which the geodesics are straight lines) from admissible


symplectic forms on G(Rn). The rest of the paper is dedicated to provingthat these projective metrics are actually Finsler metrics. A key step in theproof is the following result:

Theorem 3.5. If ω is an admissible symplectic form on G(Rn), n > 2,and Π ⊂ Rn is a plane, then the pull-back of ω to the (two-dimensional)submanifold of all oriented lines lying on Π never changes sign nor vanisheson an open set.

In order to prove this theorem we introduce two classes of open subsetsof the submanifold of all oriented lines lying on Π, G(Π). The first class,which we call discs consist of all oriented lines intersecting a line segment ina given direction (see Fig. 4). Note that the set of all oriented lines passingthrough a line segment is the union of two discs that intersect in preciselytwo points. A disc is said to be small if the line segment corresponding toit lies in a set Vx as in lemma 3.1.

Figure 4.

Lemma 3.3. We can choose an orientation on G(Π) such that the integralof ω over any disc is positive.

Proof. Notice that it is enough to look at small discs. Indeed, any line seg-ment can be partitioned into arbitrarily small line segments and the integralof ω over the original disc equals the sum of the integrals of ω over the smalldiscs corresponding to these segments.

By the symplectic interpretation of the distance d given in the proof ofproposition 3.3, it is clear that the integral of ω over a disc correspondingto a sufficiently small segment xy is, up to a sign, the quantity 2d(x, y), andthus different from zero.

We now need to see that for a given orientation the integral of ω over anytwo sufficiently small discs has the same sign. For this consider the compactset of all oriented lines on Π passing through a convex body that containsthe two segments xy and uv corresponding to the discs. The convex bodycan be finitely covered by open subsets where the symplectic interpretationof the metric d holds. Notice that we can isotopy the segment xy into thesegment uv through segments that lie inside some open set of the covering.It follows from the argument above that the integral of ω over these segmentsis never zero and, thus, they must all have the same sign. We choose theorientation so that the sign is positive and we’re done. ¤


From now on G(Π) is oriented so that the integral of ω over all discs ispositive. Unfortunately, positivity over discs is not enough to prove theo-rem 3.5. In fact, discs in G(Π) are relatively large sets: not every open subsetof G(Π) contains a disc. To remedy the situation, we introduce triangles:sets of lines intersecting two sides of a given triangle in Π in a given direc-tion (see Fig. 5). A triangle in G(Π) is said to be small if its correspondingtriangle in Π lies in a set Vx as in lemma 3.1.

Figure 5.

Lemma 3.4. The pull-back of ω to G(Π) never changes sign nor vanishesin an open set if and only if the integral of ω over any small triangle isdifferent from zero.

Proof. One part of the proof is easy: if there exists a triangle over which theintegral of ω is zero, then either ω changes sign or is identically zero insidethe triangle. On the other hand if ω changes sign or is identically zero onan open set, then we can construct a triangle over which the integral of ωis zero: if ω changes sign we can find two small triangles T+ and T− suchthat ω is positive on T+ and negative on T−. Reasoning like in the proofof the previous lemma, we can join T+ to T− by a continuous path of smalltriangles. By continuity, the integral of ω over one of those triangles mustbe zero. If ω vanishes on an open set, just pick a small triangle inside it. ¤

The following lemma concludes the proof of theorem 3.5.

Lemma 3.5. If an admissible symplectic form ω is odd, then the integralover any sufficiently small triangle on Π is positive.

Proof. Consider the four triangles T1, T2, T3, and T4 in figure 6.

T1 T2 T3 T4

z x

y

z x

y

z x

y

z x

y

Figure 6.


Notice that

2d(x, y) =∫

T2

ω +∫

T4

ω,

2d(y, z) =∫

T1

ω +∫

T3

ω,

2d(x, z) =∫

T3

ω +∫

T4

ω.

Using the strict triangle inequality for d, we have that

0 < 2d(x, y) + 2d(y, z)− 2d(x, z) =∫

T1

ω +∫

T2

ω

= 2∫

T1

ω,

where in the last equality we made use of the fact that ω is odd and that thetriangle T2 is obtained from T1 by reversing the orientation of the lines. ¤

4. Reduction to two dimensions

In this section we study the relation between the general construction ofsection 3 and the solution of Hilbert’s fourth problem in two dimensions byBusemann, Pogorelov, Ambartzumian, and Alexander (see [5] for referencesand a detailed account of their work).

A useful characterization of projective metrics on Rn that will be usedthroughout this section is provided by the following easy result:

Proposition 4.1. A continuous distance function d defines a projectivemetric on Rn if and only if

(1) If x, y, and z are collinear points with y contained in the segmentxz, then d(x, z) = d(x, y) + d(y, z).

(2) If x, y, and z are not collinear, then d(x, z) < d(x, y) + d(y, z).

In fact, condition (1) implies that straight lines are geodesics, while con-dition (2) implies that any geodesic must be a straight line.

It follows at once from this proposition that a metric on Rn is projectiveif and only if its restriction to any plane is projective. We do not evenneed to assume that planes are totally geodesic. This suggests that thekey to understanding Hilbert’s fourth problem is to understand the two-dimensional case.

The Busemann construction in two dimensions

Let µ be a (nonnegative) Borel measure defined on the space of orientedlines in R2, G(R2), that satisfies the following conditions:

(1) The set of all oriented lines passing through an arbitrary point hasmeasure zero.

(2) The measure is even: the involution a : G(R2) → G(R2) that reversesthe orientation of lines is measure-preserving.


(3) The measure of any open set is positive.Using µ we define the distance dµ(x, y) between two points x and y as

the measure of the set of all oriented lines intersecting the segment thatjoins them. The following theorems, which represent the complete solutionof Hilbert’s fourth problem in two-dimensions, is due to the joint efforts ofBusemann, Pogorelov, Ambartzumian, and Alexander (see [5] and [29] forreferences and proofs).

Theorem 4.1. The function dµ defines a projective metric on the plane.Conversely, given a projective metric d on the plane, there exists a uniquemeasure µ satisfying (1), (2), and (3) for which d = dµ.

Theorem 4.2 (Pogorelov, [27]). The function dµ is the distance functionof a projective Finsler metric on the plane if and only if µ is smooth andpositive. Moreover, every projective Finsler metric can be constructed thisway.

Reduction to two dimensionsIn order to construct a projective metric on Rn we may start by construct-

ing projective metrics on planes and imposing the following compatibilitycondition: if xy is a line segment and Π1 and Π2 are two planes containingxy, each provided with a projective metric, then the length of xy is the samewhen considered as a segment in Π1 or as a segment in Π2.

Since projective metrics on a two-dimensional subspace are in one-to-one correspondence with a class of measures on the set of lines lying onthe subspace, we look for mechanisms for inducing measures on the set oflines lying on two-dimensional subspaces. One such mechanism is to take adifferential two-form ω on the space of oriented lines in Rn and to considerits pull-backs to the manifolds of oriented lines lying on two-dimensionalsubspaces.

Using ω, we have the following variation of the two-dimensional construc-tion: if x and y are two points in Rn, and Π is a two-dimensional affinesubspace containing them, set dω(x, y; Π) to be the integral of |ω| over theset of all oriented lines on Π intersecting the segment xy.

For certain choices of ω the compatibility condition is immediately satis-fied:

Proposition 4.2. If ω is an admissible symplectic form on the manifoldof oriented lines in Rn, n > 2, then the function dω is independent of thechoice of plane and defines a projective metric on Rn.

Proof. Let Π be a two-dimensional subspace containing the segment xy.The set of all oriented lines lying on Π is a cylinder, G(Π), and the subsetof G(Π) consisting of all oriented lines intersecting xy is the union of twodiscs that intersect in a pair of points (the two oriented lines joining x andy) (Fig. 7).


Figure 7.

Each of the discs is bounded by a closed curve made up of the concate-nation of two simple curves. One of this simple curves lies in x, the set ofall oriented lines passing through x, the other in y. They both join the twointersection points of x and y. Using Stokes’ formula and the admissibilityof ω, it is easy to prove the following statement:

Let σ be any closed curve in the space of oriented lines in Rn that ismade up of the concatenation of two simple curves, one in x, the other y,and both joining the points in x∩ y. If D is any disc with boundary σ, thenthe absolute value of the integral of ω over D depends only on x and y.

Since, by theorem 3.5 the pull-back of ω to the submanifold of orientedlines lying Π never changes sign, we have that

dω(x, y, Π) = 2∫

D|ω| = 2

∣∣∣∣∫

Dω

∣∣∣∣ ,

where D is any one of the two discs in figure 7. This implies that dω isindependent of the choice of two-dimensional subspace.

To see that dω defines a projective metric, we note that on each plane Πthe measure |ω| defined on G(Π) is continuous and even. In particular, itsatisfies conditions (1) and (2) of the Busemann construction. Moreover, bytheorem 3.5, |ω| is positive on open sets (condition (3)) and thus the restric-tion of dω to Π is the metric one obtains from Busemann’s two-dimensionalconstruction. ¤

Note that we now have two different ways to construct a projective metricin Rn from an admissible symplectic form on G(Rn). The first is to use theconstruction in section 3 and the second is given by the previous proposition.Since the metrics d and dω/4 are intrinsic and agree locally (i.e., they arecomputed in the same way from the symplectic area of certain discs), theyare the same. However, in the second construction it is easier to see how tobridge the gap between G-space metrics and Finsler metrics:

Proposition 4.3. Let ω be an admissible two-form on the manifold of ori-ented lines in Rn, n > 2. If the pull-back of ω to to the submanifold of alloriented lines lying on an arbitrary plane Π never vanishes, the function dω

defines a projective Finsler metric.


Proof. In the proof of proposition 4.2, we only used the fact that ω was sym-plectic in order to apply theorem 3.5. Since we are now assuming somethingstronger, the proof carries on without modification to show that dω definesa projective metric on Rn. Moreover, by theorem 4.2, the restriction of thismetric to every plane is a Finsler metric. This immediately implies that themetric is Finsler. ¤

It is also possible to show that if dω is a Finsler metric, the form ω is thenatural symplectic form in its space of geodesics. However, this needs thedeeper study of admissible two-forms in the following section.

5. A characterization of admissible symplectic forms

In this section we shall prove the following result, thus completing theproof of our main theorem.

Theorem 5.1. An admissible two-form on the manifold of oriented linespassing through an open convex subset M ⊂ RPn is symplectic if and onlyif for every plane Π intersecting M the submanifold of all oriented linespassing through M and lying on Π is symplectic.

In order to prove this result, we must undertake a detailed study of thegeneralized conformal structure of the Grassmannian of 2-planes in Rn+1,G+

2 (Rn+1):Recall that the tangent space to G+

2 (Rn+1) at a plane P can be identifiedwith the space of linear transformation from P to its orthogonal complementP⊥. In fact, a whole neighborhood of P in G+

2 (Rn+1) can be identified withthis vector space since any plane sufficiently close to P is the graph inP ⊕ P⊥ = Rn+1 of some linear transformation from P to P⊥.

Definition 5.1. The generalized conformal structure on the GrassmannianG+

2 (Rn+1) assigns to every plane P ∈ G+2 (Rn+1) the incidence cone in

TP G+2 (Rn+1) that consists of all rank-one linear transformations from P

to P⊥.

The incidence cones are made up of two types of linear subspaces, theα-planes and the β-planes. We start by describing the α-planes.

Choosing a basis of P and a basis of P⊥ we represent the tangent spaceTP G+

2 (Rn+1) of 2× (n− 1) matrices(

q1 . . . qn−1

p1 . . . pn−1

).

Given real numbers λ1 and λ2 we define the n-dimensional subspace

H(λ1,λ2) :={(

λ1q1 . . . λ1qn−1

λ2q1 . . . λ2qn−1

): q1, . . . , qn−1 ∈ R

}.

Note that H(λ1,λ2) depends only on the quotient of λ1 and λ2. In otherwords the H(λ1,λ2) define a projective line of n-dimensional subspaces in


TP G+2 (Rn+1). Since all the matrices in H(λ1,λ2) have rank one, the subspaces

are all inside the incidence cone.The description of the β-planes is equally simple: for real numbers λ1, . . . , λn−1

define the 2-dimensional subspace

K(λ1,...λn−1) :={(

λ1q . . . λn−1qλ1p . . . λn−1p

): q, p ∈ R

}.

Note that the set of β-planes in TP G+2 (Rn+1) is parameterized by a projec-

tive space of dimension n− 2.An alternate, projective, description of α-planes and β-planes is as follows:

if x is a point in RPn and l is an oriented line passing through x, then Tlxis an α-plane, and all α-plane are of this form. Similarly, if Π ⊂ RPn is atwo-dimensional subspace and l is an oriented line lying on Π, then TlG(Π)is a β-plane and all β-planes are of this form.

Note that we can now characterize admissible two-forms on the space oforiented lines passing through a open convex subset of projective space as adifferential two-form that is odd, closed, and that vanishes on every α-plane.

Two-forms vanishing on α-planes

We shall now describe all two-forms ω ∈ Λ2(TP G+2 (Rn+1)) which vanish

on the subspaces of the form H(λ1,λ2), λ1, λ2 ∈ R. In the computations thatfollow we identify the space of 2× (n− 1) matrices with R2n−2 by using thecoordinate functions q1, . . . , qn−1, p1, . . . , pn−1 as we have been doing in theprevious paragraphs.

Proposition 5.1. A 2-form ω ∈ Λ2(TP G+2 (Rn+1)) vanishes on all α-planes

if and only it is of the form ω =∑

bijdqi ∧ dpj with bij = bji. In particular,such ω is non-degenerate if and only if the matrix (bij) is invertible.

Proof. A general 2-form ω ∈ Λ2(TP G+2 (Rn+1)) has the form

ω =∑

aijdqi ∧ dqj +∑

bijdqi ∧ dpj +∑

cijdpi ∧ dpj ,

with aij = −aji and cij = −cji. If ω vanishes on the subspaces of the form

H(λ1,0) :={(

λ1q1 . . . λ1qn−1

0 . . . 0

): q1, . . . , qn−1 ∈ R

},

then∑

aijdqi ∧ dqj = 0 and, therefore, the coefficients aij are all equal tozero. Similarly, if ω vanishes on the subspaces of the form H(0,λ2), then thecoefficients cij are all equal to zero.

Finally, if ω is to vanish on a subspace of the form H(λ1,λ2) with neither λ1

nor λ2 equal to zero, then∑

bijdqi∧dpj = 0. This implies that bij = bji. ¤

We shall denote the subspace of Λ2(TP G+2 (Rn+1)) consisting of those 2-

forms which vanish on all α-planes by Λ−P . The previous proposition showsthat the dimension of this subspace equals n(n − 1)/2. The bundle overG+

2 (Rn+1) whose fiber over every point P is Λ−P will be denoted by Λ−.


Clearly, an admissible 2-form is a section of this bundle. If the admissible2-form is symplectic, then it is a nowhere-vanishing section.

We shall now prove that an admissible 2-form which doesn’t vanish onβ-planes is necessarily symplectic. This proves the first (easier) part oftheorem 5.1.

Proposition 5.2. The form ω :=∑

bijdqi ∧ dpj ∈ Λ−P never vanishes on aβ-plane if and only if the matrix (bij) is definite. Therefore, if an admissible2-form never vanishes on a β-plane, it is symplectic.

Proof. Let us evaluate the form ω on a basis of the β-plane K(λ1,...,λn−1).

ω

((λ1q . . . λn−1qλ1p . . . λn−1p

),

(λ1q

′ . . . λn−1q′

λ1p′ . . . λn−1p

′

))=

∑bij(λiqλjp

′ − λipλjq′)

= (qp′ − pq′)∑

bijλiλj .

Since the vectors are linearly independent the quantity qp′ − pq′ is not zeroand the quantity

∑bijλiλj is nonzero for all nonzero values of (λ1, . . . , λn−1)

if and only if the matrix (bij) is definite. ¤

Notice that the previous proof also implies the following result:

Corollary 5.1. Two 2-forms on Λ−P that agree on every β-plane in TP G+2 (Rn+1)

are equal. Therefore, two admissible 2-forms that agree on every β-plane areequal.

The signature of the matrix (bij) does not depend on the choice of baseson the plane P and its orthogonal complement P⊥. In particular if ω is anon-degenerate 2-form that is also a section of Λ− over some connected opensubset U ⊂ G2(Rn+1), then the signature of the matrix (bij) is the same atall points P ∈ U . It is thus natural to propose the following definition:

Definition 5.2. Let ω be a non-degenerate 2-form that is also a section ofΛ− over a connected open subset U ⊂ G2(Rn+1). The signature of ω is thesignature of the matrix (bij) at any point P ∈ U . We shall call the 2-formω definite if the matrix (bij) is definite and indefinite otherwise.

It is easy to construct admissible symplectic forms on open subsets ofG2(Rn+1) that are not definite. However, this is not possible if the opensubset is connected and contains the set of all lines passing through a givenpoint in RPn.

Theorem 5.2. Let U ⊂ G2(Rn+1) be a connected open subset that containsthe set of all oriented lines passing through a point x ∈ RPn. If ω is anadmissible symplectic 2-form defined on U , then ω is definite.

Proof. Without loss of generality, we may assume that U is the set of ori-ented lines passing through an open convex subset M of RPn. The strategyof the proof consists in showing that if ω is indefinite, then there exists aplane Π passing through M and such that the pull-back of ω to the manifold


of all lines passing through M and lying on Π either changes sign or vanisheson an open set. According to theorem 3.5 this cannot be the case if ω issymplectic.

Consider the double fibration

Fπ1

ÄÄÄÄÄÄ

ÄÄÄÄ π2

ÂÂ@@@

@@@@

U B,

where B is the set of all projective planes in RPn that contain a line in Uand F denotes the incidence relation {(l,Π) ∈ U × B : l ⊂ Π}. Remarkthat B is an open subset of the Grassmannian of projective planes in RPn,G3(Rn+1).

If ω is an indefinite 2-form on U , we define N ⊂ F to be the set of allpairs (l,Π) such that ω vanishes on the β-plane TlG(Π). From the proofof proposition 5.2, it follows that N is a smooth hypersurface and thatN ∩ π−1(l) is a non-degenerate projective quadric for each l ∈ U .

Notice that if Π ∈ B and π−12 (Π) intersects N transversally at some point,

then the pull-back of ω to G(Π) ∩ U changes sign. If π−12 (Π) is contained

in N , then the pull-back of ω vanishes on G(Π) ∩ U . Therefore, in order toprove the theorem, it is enough to show that if no fiber of π2 intersects Ntransversally, then there exists at least one that is wholly contained in N .

If no fiber is transverse, we may consider the distribution of tangent planeson N given by (l, Π) 7→ T(l,Π)π

−12 (Π). Since this distribution is integrable in

F ⊃ N , it is integrable in N and its leaves are the fibers of π2. It followsthat there is a fiber of π2 that lies completely on N and this finishes theproof. ¤

Proof of theorem 5.1. Let M be an open convex subset of RPn, n > 2, andlet ω be an admissible symplectic form on G(M). By the previous theoremω is a definite form. Applying proposition 5.2 we have that ω never vanisheson a β-plane and, therefore, the pull-back of ω to the submanifold of orientedlines lying on a plane that passes through M never vanishes.

On the other hand if ω is an admissible two-form on G(M) such thatits pull-back to the submanifold of oriented lines lying on any plane thatpasses through M never vanishes, then it never vanishes on a β-plane, andby proposition 5.2, it is symplectic. ¤

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J.C. Alvarez Paiva, Polytechnic University, 6 MetroTech Center, Brook-lyn, New York, 11201.

E-mail address: [email protected]

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