Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 39 (1998), No. 1, 75-83.

Some Affine Properties of the k-Symplectic Manifolds

A. Awane

Université Hassan II - Mohammedia Faculté des Sciences II, Département de Mathématiques B.P. 7955 Ben Msik, Casablanca. Maroc

Abstract. We study some affine properties of the k-symplectic manifolds and also the complete locally affine k-symplectic manifolds; a familly of these spaces is given in the compact 3-dimensional case.

MSC 1991: 58F05, 53C15

Keywords: Lagrangian foliations, k-symplectic structure, locally affine manifolds.

1. Introduction

Mathematical and physical considerations (the local study of Pfaffian systems and Nambu’s statistical mechanic) have led us to introduce the notion of k-symplectic structure [4]. By the Heisenberg group of rank k in the sense of Goze-Haraguchi [11] we see that the k-symplectic geometry is related to the k-contact systems in analogy with the well-known relationship between symplectic and contact structures [6].

The Kostant-Souriau’s geometric prequantization is obtained in the context of these structures [15].

A k-symplectic structure on an n(k + 1)-dimensional manifold M is given by an ncodimensional foliation F and a system of k differential closed 2-forms vanishing on the subbundle of T M defined by F with transversal characteristic spaces. The fundamental example is given by the natural k-symplectic structure on a fibre product of k Lagrangian fibrations, in particular we have a k-symplectic structure on the Whitney sum T^{∗}B ⊕ … ⊕ T^{∗}B over an n-dimensional manifold B [5].

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1998 Heldermann Verlag

76 A. Awane: Some affine properies of the k-symplectic manifolds

In terms of G-structures, a k-symplectic structure on M can be defined by an integrable Sp(k, n; ℝ)-structure, where Sp(k, n; ℝ) is the group of all matrices of the type

( \begin{matrix}T & & 0 & S_{1} \\ & ... & & ⋮ \\ & & T & S_{k} \\ 0 & & & ^{t}(T^{−1}) \end{matrix} )

where T ∈ Gl(n, ℝ) , S_{1}, ⋯, S_{k} are n×n real matrices such that T ^{t}S_{α} = S_{α} ^{t}T.

The particular case k = 1 corresponds to a classical symplectic structure endowed with a Lagrangian foliation. This last structure is usually called a real polarization of a symplectic manifold or a quasi-cotangent structure in the sense of Molino, Clark and Goel. The generalization to the k-system is strongly motivated by the geometrical formalization of the statistical mechanic.

This paper is devoted to the study of foliations subordinate to a k-symplectic structure and locally affine manifolds endowed with a k-symplectic structure. We study some affine properties of k-symplectic manifolds; in particular we show that a k-symplectic manifold M is locally affine relatively to the nk-dimensional differential structure on M given by the leaves of the foliation F . This leads us to study some properties of the complete connected locally affine k-symplectic manifolds; these manifolds are quotients ℝ^{n(k+1)}/Γ where Γ are subgroups of the group of all affine transformations X ⟼ AX + B of ℝ^{n(k+1)} such that A ∈ Sp(k, n; ℝ) , acting freely and properly discontinuously on ℝ^{n(k+1)} , and also, we give an infinite family in the 3-dimensional compact case.

2. Affine structure of the characteristic foliations

Let M be an n(k+1)-dimensional smooth manifold. Let E be an integrable n-codimensional subbundle of T M . Let θ^{1}, …, θ^{k} be closed 2-forms on M . Let T M/E be the quotient bundle. Let ν be the canonical projection T M ⟶ T M/E = νE. Let ν^{∗}E be the dual of νE. We denote by F the foliation defined by the subbundle E. For each α, (α = 1 , …, k) and x ∈ M , we denote by C_{x}(θ^{α}) the characteristic space of θ^{α} at x.

All geometric objects on M are assumed to be of class C^{∞}.

We say that (θ^{1}, …, θ^{k}; E) defines a k-symplectic structure on M if for each x ∈ M , the following conditions are satisfied :

1) C_{x}(θ^{1}) ∩ ⋯ ∩ C_{x}(θ^{k}) = {0},

2) θ^{α}(X, Y) = 0 for all X, Y ∈ Γ(E) and α, (α = 1 , …, k). A k-symplectic manifold is a manifold with a k-symplectic structure.

Example 2.1. A k-symplectic structure on the cotangent bundle of k^{1} − covelocities. Let B be an n-dimensional manifold. We denote by T_{k^{1}}^{∗}B the cotangent bundle of k^{1} − covelocities of B, that is the manifold of all 1-jets of mapping from B to ℝ^{k} with target 0 ∈ ℝ^{k}.

For each coordinate system (x^{j})_{1≤j≤n} on B we associate the local coordinates

(x^{i}, x_{i}^{1}, …, x_{i}^{k})_{1≤i≤n}

on T_{k^{1}}^{∗}B defined by x^{i}(J_{x,0^{1}}f) = x^{i}(x)

x_{i}^{α}(J_{x,0^{1}}f) = (\frac{∂f^{α}}{∂x^{i}})(x),

where J_{x,0^{1}}f is the 1-jet at x ∈ B of the map f = (f^{1}, …, f^{k}) : B ⟶ ℝ^{k} such that f(x) = 0 .

We have T_{k^{1}}^{∗}B is an n(k + 1)-dimensional vector bundle with standard fibre type ℝ^{nk}; the canonical projection is the map π : T_{k^{1}}^{∗}B ⟶ B defined by π(J_{x,0^{1}}f) = x.

For each α, (α = 1 , …, k) we have a canonical 1-form λ^{α} defined by

λ^{α}(u)(X) = u^{α}(x)(π_{∗}X)

where u = (u^{1}, …, u^{k}) ∈ T_{k^{1}}^{∗}B, X ∈ T_{u}(T_{k^{1}}^{∗}B) and π(u) = x.

Let θ^{α} = dλ^{α} where α = 1, …, k and let E = Ker π_{∗}. The (k+1)-tuple (θ^{1}, …, θ^{k}; E) defines a k-symplectic structure on T_{k^{1}}^{∗}B.

Let (θ^{1}, …, θ^{k}; E) be a k-symplectic structure on M . Recall [4] (see also [5]) that about every point of M , we can find an open neighborhood U and local coordinates (x^{αi}, x^{i})_{1≤α≤k,1≤i≤n} on U , called adapted coordinates, so that

θ_{|U}^{α} = ∑_{i=1}^{n} dx^{αi} ∧ dx^{i} , E_{|U} = Ker dx^{1} ∩ … ∩ Ker dx^{n}

for each α, (α = 1 , …, k) (Darboux theorem).

Equivalently, there exists an atlas A of M, called Darboux atlas, such that the changes of coordinates are in the pseudogroup of local diffeomorphisms of ℝ^{n(k+1)} leaving the canonical k-symplectic structure invariant. It follows from the previous canonical Darboux expressions that the differential forms θ^{α}, (α = 1, …, k) are of class 2n; this implies that the distribution x ⟼ C_{x}(θ^{α}) defines integrable subbundles of T M .

For each α, (α = 1, …, k) we take

E^{α} = ⋂_{β≠α} C(θ^{β}).

In terms of adapted coordinates (x^{αi}, x^{i})_{1≤α≤k,1≤i≤n}, νE is spanned by the derivatives \frac{∂}{∂x^{1}} , …, \frac{∂}{∂x^{n}} , ν∗E is spanned by the differential forms dx^{1}, …, dx^{n} and E^{α} is spanned by the derivatives \frac{∂}{∂x^{α1}} , …, \frac{∂}{∂x^{αn}} .

We have :

i) for each α, (α = 1, …, k), the subbundle E^{α} is integrable, ii) E = E^{1} ⊕ … ⊕ E^{k} (direct sum), iii) for each α, (α = 1, …, k) the map X ⟼ i(X)θ^{α} defines an isomorphism i_{α} of vector

bundles over M of E^{α} onto ν^{∗}E.

Let F^{α} be the foliation of M defined by the subbundle E^{α}, (α = 1, …, k). We call F^{1}, …, F^{k} the characteristic foliations of the k-symplectic structure.

In terms of adapted coordinates (x^{αi}, x^{i})_{1≤α≤k,1≤i≤n} the map i_{α} expresses the duality \frac{∂}{∂x^{αs}} ⟼ dx^{s} between the geometry along the leaves of F^{α} and the transverse geometry.

The Darboux expressions imply the following local transition formulas of the canonical coordinates

x̄^{αi} = ∑_{j=1}^{n} \frac{∂x^{j}}{∂x̄^{i}} x^{αj} + φ^{αi}(x^{1}, …, x^{n}) , x̄^{i} = x̄^{i}(x^{1}, …, x^{n}) ,

where φ^{αi} are some well defined functions. Indeed, these expressions are affine with respect to x^{αi}; this proves that we have

Proposition 2.2. ^{1} Each leaf of F^{α}(α = 1, …, k) (resp. F ) is locally affine.

Invariantly, the affine structure of the leaves under discussion has as local parallel vector fields the fields which correspond to the E-projectable 1-forms under the isomorphism i_{α}.

We give now a flat affine connection along the leaves. Let \overline{▽} be the basic connection on νE [8]; it is the ℝ -bilinear map \overline{▽} : Γ(E)×Γ(νE) ⟶ Γ(νE) given by \overline{▽}_{X}Z = ν[X, \overline{Z}] , for each X∈Γ(E) and Z∈Γ(νE) where \overline{Z} ∈ χ(M) such that ν(\overline{Z}) = Z.

Recall that the curvature k̄ of \overline{▽} is a correspondence that associates to every pair (X, Y) ∈ Γ(E) × Γ(E) a mapping k̄(X, Y) : Γ(νE) ⟶ Γ(νE) given by

k̄(X, Y)Z = \overline{▽}_{X} \overline{▽}_{Y} Z − \overline{▽}_{Y} \overline{▽}_{X}Z − \overline{▽}_{[X,Y]}Z, Z ∈ Γ(νE) .

We have k̄(X, Y) = 0 for each X , Y ∈ Γ(E).

Let α = 1, …, k and let L be a leaf of F^{α}. For each X, Y ∈ χ(L) we take

▽_{X}Y = i_{α}^{−1}(\overline{▽}_{X}^{∗}(i_{α}(Y)))

where \overline{▽}^{∗} is the map of Γ(E^{α}) × Γ(ν^{∗}E) into Γ(ν^{∗}E) defined by

(\overline{▽}_{X}^{∗}σ)Z = L_{X}(σ(Z)) − σ(\overline{▽}_{X}Z)

for each X ∈ Γ(E^{α}) , σ ∈ Γ(ν^{∗}E) and Z ∈ Γ(νE) ; L_{X} denotes the Lie derivative with respect to X. ▽ defines an affine connection along the leaf L.

Let T be the torsion tensor of ▽. Let Z be a local cross section of the bundle νE ⟶ M and let \overline{Z} be a local cross section of the bundle T M ⟶ M such that ν(\overline{Z}) = Z. We have

θ^{α}(T (X, Y), \overline{Z}) = θ^{α}(▽_{X}Y − ▽_{Y} X, \overline{Z}) − θ^{α}([X, Y], \overline{Z})

and

\begin{aligned}θ^{α}(▽_{X}Y − ▽_{Y} X, \overline{Z}) & = θ^{α}(i_{α}^{−1}(\overline{▽}_{X}^{∗}(i_{α}(Y))\overline{▽}_{Y}^{∗} (i_{α}(X))) , \overline{Z}) \\ & = (\overline{▽}_{X}^{∗}(i_{α}(Y)))(Z) − (\overline{▽}_{Y}^{∗} (i_{α}(X)))(Z), \end{aligned}

1The proof of this proposition was indicated to me by the referee

then

\begin{aligned}θ^{α}(▽_{X}Y − ▽_{Y} X, \overline{Z}) & = L_{X}i_{α}(Y)(Z) − i_{α}(Y)\overline{▽}_{X}Z − L_{Y} i_{α}(X)(Z) + i_{α}(X)\overline{▽}_{Y} Z \\ & = L_{X}(θ^{α}(Y, \overline{Z})) − θ^{α}(Y, [X, \overline{Z}]) + L_{Y} (θ^{α}(\overline{Z}, X)) + θ^{α}(X, [Y, \overline{Z}]) \end{aligned}

and also

θ^{α}(T (X, Y), \overline{Z}) = L_{X}(θ^{α}(Y, \overline{Z})) + L_{Y} (θ^{α}(\overline{Z}, X))
+ θ^{α}(X, [Y, \overline{Z}]) − θ^{α}(Y, [X, \overline{Z}]) − θ^{α}([X, Y], \overline{Z})
= dθ^{α}(X, Y, \overline{Z}).

θ^{α} is closed, Z is arbitrary and i_{α} is an isomorphism, then T (X, Y) = 0 .

The curvatures k, k̄ and k̄^{∗} of ▽, \overline{▽} and \overline{▽}^{∗} respectively are related by

i_{α}(k(X_{1}, X_{2})Y) = k̄^{∗}(X_{1}, X_{2})(i_{α}(Y))

where X_{1} , X_{2} , Y ∈ χ(L) .

We have k̄ ≡ 0 , then k̄^{∗} ≡ 0 and k ≡ 0 ; that is ▽ is a flat affine connection.

As above, we define the flat connection along the leaves of the foliation F as the connection which induces the other flat connections along the leaves of the characteristic foliations.

Let M be a k-symplectic manifold. If M is compact then the fundamental group π_{1}(M) of M is infinite, and if H_{1}(M, ℤ) = 0 then the foliation F is defined by a nowhere zero closed n-form on M [7]. quad Recall [5] that an infinitesimal automorphism X of the subbundle E is a Hamiltonian system if the differential forms i(X)θ^{1}, …, i(X)θ^{k} are closed.

Let X be a Hamiltonian system of a k-symplectic structure (θ^{1}, …, θ^{k}; E) . In terms of adapted local coordinates (x^{αi}, x^{i})_{1≤α≤k,1≤i≤n}, the vector field X takes the form

∑_{i=1}^{n} ( − ∑_{α=1}^{k}(∑_{j=1}^{n} \frac{∂f_{j}}{∂x^{i}} (x^{1}, …, x^{n})x^{αi} + \frac{∂g^{α}}{∂x^{i}} (x^{1}, …, x^{n}))\frac{∂}{∂x^{αi}} + f_{i}(x^{1}, …, x^{n})\frac{∂}{∂x^{i}} )

where f_{j} and g^{α} are smooth functions. Then the flow generated by X respects the affine structure of the leaves of F, i.e. X is an infinitesimal affine transformation of the space M with its differential structure of dimension nk defined by the leaves of F.

3. Locally affine k-symplectic manifolds

Let M be an n(k + 1)-dimensional manifold with a k-symplectic structure

(θ^{1}, …, θ^{k}; E).

M is called locally affine if the Darboux atlas A confers to M a structure of locally affine manifold.

Let Gp(k, n; ℝ) be the group of all affine transformations of ℝ^{n(k+1)} leaving the canonical k-symplectic structure of ℝ^{n(k+1)} invariant. The group Gp(k, n; ℝ) is the set of all affine transformations X ⟼ AX + B of ℝ^{n(k+1)} such that A ∈ Sp(k, n; ℝ) .

Let M be a complete connected locally affine k-symplectic manifold of dimension n(k+ 1) . By [2] and [3] (see also [17]), the manifold M is just a quotient ℝ^{n(k+1)}/Γ and with a fundamental group Γ , where Γ is a subgroup of A(n(k + 1)) acting freely and properly discontinuously on ℝ^{n(k+1)} .

Let q : ℝ^{n(k+1)} ⟶ M = ℝ^{n(k+1)}/Γ be the canonical projection ( q is covering map), and let P be the subbundle of T ℝ^{n(k+1)} defined by q_{∗}P_{x} = E_{q(x)} for all x ∈ ℝ^{n(k+1)} . The (k + 1)-tuple (q^{∗}θ^{1}, …, q^{∗}θ^{1}; P) defines a Γ-invariant k-symplectic structure on ℝ^{n(k+1)}. The changes of coordinates of M are affine transformations of ℝ^{n(k+1)} leaving the canonical k-symplectic structure invariant. By the definition of the quotient M = ℝ^{n(k+1)}/Γ , we see that Γ is contained in Gp(k, n; ℝ) .

Let Hp(k, n; ℝ) be the group of all matrices

( \begin{matrix}I_{n} & & 0 & S_{1} & T_{1} \\ & ... & & ⋮ \\ & & I_{n} & S_{k} & T_{k} \\ 0 & & & I_{n} & Q \\ 0 & & & 0 & 1 \end{matrix} )  (3.1)

where I_{n} is the unit matrix of rank n, S_{1}, ⋯, S_{k} are n × n real symmetric matrices and T_{1}, …, T_{k}, Q are column vectors of rank n.

We denote by (S, Q, T) the matrices of the form (3.1) where S = (S_{1}, …, S_{k}) , T = (T_{1}, …, T_{k}).

The affine group Hp(k, n; ℝ) admits similar properties as the Heisenberg group of rank k in the sense of Goze-Haraguchi; it is a nilpotent, connected and simply connected Lie group such that [Hp(k, n; ℝ), Hp(k, n; ℝ)] coincides with the center of Hp(k, n; ℝ) .

Assume that n = 1 . An affine transformation g ∈ Gp(k, 1; ℝ) is without a fixed point if and only if the following conditions are satisfied:

(1) g = (S, q, T) ∈ Hp(k, 1; ℝ).

(2) T ∉ ℝ.S. In particular we have Γ ⊂ Hp(k, 1; ℝ).

We now assume that n = 1 and k = 2 . Let Γ_{2}^{02012} be the subgroup of Hp(2, 1, ℝ) generated by

(S^{0}, 0, T^{0}) , (mS^{0}, 0, T^{1}) , (S^{2}, q_{2}, T^{2})

where S^{0}, S^{2}, T^{0}, T^{1}, T^{2} ∈ ℤ^{2}, q_{2} ∈ IN^{∗} , m ∈ ℚ with q_{2} ≠ 0, det(T^{0}, T^{1}) ≠ 0 and δ = mD_{2} ; here δ and D_{2} denote the determinants det(T^{1}, S^{0}) and det(T^{0}, S^{0}) respectively.

Proposition 3.1. We have:

(i) The subgroup Γ_{2}^{02012} acts freely and properly discontinuously on ℝ^{3}.

(ii) The quotient M_{2}^{02012} = ℝ^{3}/Γ_{2}^{02012} is an orientable complete 2-symplectic compact connected locally affine manifold.

(iii) The manifold M_{2}^{02012} is homeomorphic to the torus T^{3} if and only if S^{0} = 0 .

Proof. (i) Every element g of Γ_{2}^{02012} can be written g = (S, q, T) with

(a) S = ( ∑_{i=1}^{n} α_{i} + m ∑_{i=1}^{n} β_{i})S^{0} + (∑_{i=1}^{n} γ_{i})S^{2}
(b) q = (∑_{i=1}^{n} γ_{i})q_{2}
(c) T = ( ∑_{p=1}^{n} ( ∑_{j=1}^{p}(α_{j} + mβ_{j}))γ_{p})q_{2}S^{0} + q_{2}P_{(∑_{i=1}^{n} γ_{i})}S^{2} + (∑_{i=1}^{n} α_{i})T^{0} +
(∑_{i=1}^{n} β_{i})T^{1} + (∑_{i=1}^{n} γ_{i})T^{2}

where n ∈ ℕ, α_{i}, β_{i}, γ_{i} ∈ ℤ for all i , j , k = 1 , …, n and P_{γ} = γ(γ − 1)/2.

The subgroup G = Γ_{2}^{02012} acts freely on ℝ^{3} . We show now that G acts properly discontinuously on ℝ^{3} . Two cases are possible.

Case 1: S^{0} ≠ 0. Let p_{0} ∈ ℤ and q_{0} ∈ IN^{∗} such that m = \frac{p_{0}}{q_{0}} . Let Q_{0} = (x_{0}, y_{0}, z_{0}) ∈ ℝ^{3}, let

ε = min(\frac{1}{2q_{0}} , \frac{|D_{2}|}{2q_{0}(|s_{1}^{0}| + |s_{2}^{0}|)}),

where S^{0} = (s_{1}^{0}, s_{2}^{0}) , and let U_{0} be the open ball B(Q_{0}, ε) with center at the point Q_{0} and with radius ε (for the norm ||(u, v, w)|| = max(|u|, |v|, |w|)).

Let g ∈ G such that gU∩U ≠ ∅ and let Q = (x, y, z) ∈ U such that g(Q) = (x′, y′, z′) ∈ U . Then we have |x − x′| < 2ε, |y − y′| < 2ε and |z − z′| < 2ε.

By (a), (b), (c) we have |z − z′| = |(∑_{i=1}^{n} γ_{i})q_{2}| < 2ε < 1 , then | ∑_{i=1}^{n} γ_{i}| is an integer less than 1, then ∑_{i=1}^{n} γ_{i} = 0 and z = z′. By the inequalities |x−x′| < 2ε and |y−y′| < 2ε we deduce that |s_{2}^{0}(x′ − x) − s_{1}^{0}(y′ − y)| ≤ 2ε(|s_{1}^{0}| + |s_{2}^{0}|) .

And also by (a) , (b) , (c) we have |( ∑_{i=1}^{n} α_{i} + m ∑_{i=1}^{n} β_{i})q_{0}| < 1 , then ∑_{i=1}^{n} α_{i} + m ∑_{i=1}^{n} β_{i} = 0. It follows that q_{0}|x − x′| and q_{0}|y − y′| are two integers less than 1, then x = x′, y = y′ and also g = (0, 0, 0) ; in other words, we have

{g ∈ G / gU∩U ≠ ∅} = {idℝ^{3}}.

Consider now two points Q_{1} and Q_{2} of ℝ^{3} not congruent modulo G, let K be a compact set contained in ℝ^{3} and let r > 0 such that K ⊂ B(Q_{1}, r) , where B(Q_{1}, r) is the open ball with center at the point Q_{1} and with radius r.

The following sets

E_{1} = {γ ∈ ℤ / |γq_{2}| < r}
E_{2} = { (α + mβ)q_{0} / |α + mβ| < 2r(|s_{1}^{0}| + |s_{2}^{0}|) , α , β ∈ ℤ}

and E_{(j,3)} = { (α + mβ)z_{1}s_{j}^{0} + γt_{j}^{2} + u / |(α + mβ)z_{1}s_{j}^{0} + γt_{j}^{2} + u| < r ,

(α + mβ)q_{0} ∈ E_{2} , γ ∈ E_{1} , u ∈ ℤ} (j = 1 , 2)

where T^{2} = (t_{1}^{2}, t_{2}^{2}) , are finite. Then G.Q_{1}∩K is finite; this implies that

inf\{ d(Q, Q_{1}) = ||\overrightarrow{QQ_{1}}|| / Q ∈ G.Q_{1}\} = α > 0.

Let

ε′ = min(\frac{1}{q_{0}} , \frac{|D_{2}|}{4q_{0}(|s_{1}^{0}| + |s_{2}^{0}|)} , \frac{α}{2}) ,

and let U_{1} (resp. U_{2} ) be the open ball with center Q_{1} (resp. Q_{2} ) and with radius ε′. We have

G.U_{1}∩U_{2} = ∅.

Case 2: S^{0} = 0. We may prove the assertion for this case exactly as for the first case. Observe that, in this case, the group Γ_{2}^{02012} is the free abelian group on three generators.

(ii) The fact that the vectors (T^{0}, 0) , (T^{1}, 0) and (T^{2}, q_{2}) are linearly independent permits us to see that the quotient M_{2}^{02012} is a compact manifold.

The orientability of this manifold follows from the fact that each element of the group Hp(2, 1, ℝ) has positive determinant.

(iii) By [2] and [3], two compact connected complete locally affine manifolds M_{1} and M_{2} are homeomorphic if and only if the fundamental groups π_{1}(M_{1}) and π_{1}(M_{2}) are isomorphic; this implies that M_{2}^{02012} is homeomorphic to the torus T^{3} if and only if the fundamental group π_{1}(M_{2}^{02012}) = Γ_{2}^{02012} of M is an abelian group if and only if S^{0} = 0.

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Received June 7, 1996; revised version November 1, 1996