Journal of Lie Theory Volume 11 (2001) 9–22 ◯^{C}2001 Heldermann Verlag

Reduction in contact geometry

Frank Loose

Communicated by K.-H. Neeb

Abstract. The notion of a moment map for (exact) contact manifolds is introduced. We prove a reduction theorem analogous to the Marsden-Weinstein theorem in the (exact) symplectic case. Furthermore the existence of a local normal form for the moment map in a neighborhood of a point in the zero moment level is proved. This implies a local slice theorem for contact transformation groups.

1. Introduction and statement of results

Consider a smooth manifold C. A distribution H ⊆ T C of hyperplanes in the tangent bundle π: T C → C induces a bundle morphism ω: H × H → L, where H × H denotes the bundle product {(ξ, η) | π(ξ) = π(η)} and L is the quotient bundle T C/H, given by the Lie bracket on the vector fields on C. In fact, when two vectors ξ and η in the fibre H_{c} over c ∈ C are extended to vector fields X and Y respectively on C which are sections of H, then the class of [X, Y]_{c} in T C_{c}/H_{c} = L_{c} does not depend on the choice of the extensions; thus ω_{c}(ξ, η) = [X, Y]_{c} mod H_{c} ∈ L_{c} is well-defined. We say that the distribution H ⊆ T C is of constant rank, if the rank of the skew-symmetric bilinear map ω_{c} does not depend on c. In case that ω is of maximal rank (i.e., ω is non-degenerate) H is called a contact structure on C and (C, H) is a contact manifold.

A special situation occurs when the distribution H is given as the kernel of a globally defined differential 1-form α on C, H_{c} = ker(α_{c}). In this case we refer to the pair (C, α) as an exact contact manifold. If φ is a nowhere vanishing function on C, then the form φα clearly induces the same contact structure on C. However, the exact contact manifold (C, φα) is different from (C, α).

Let G be a Lie group. If (C, H) is a contact manifold and G acts on C by diffeomorphisms, G × C → C, (g, c) ↦ g.c, we say that G acts by contactomorphisms, if any transformation g: C → C, c ↦ g.c respects the contact structure, i.e., g_{∗}H = g^{∗}H. (We denote by g^{∗}H the pullback bundle of H, i.e., (g^{∗}H)_{c} = H_{g.c} and by g_{∗}: T C → g^{∗}T C the differential of g: C → C.) Similarly, if (C, α) is an exact contact manifold, then G is said to act by exact

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag contactomorphisms, if any g: C → C respects α, i.e., g^{♯}α = α. (We denote by g^{∗}α: g^{∗}T C → R the pullback of α as a section and by g^{♯}α = g^{∗}α ∘ g_{∗}: T C → R the pullback of α as a 1-form on C.) In analogy to the exact symplectic case (see e.g. [8]) we define the corresponding moment map as the smooth map from C into the dual of the Lie algebra g of G, Φ: C → g^{∗}, given by

Φ_{a}(c) := ⟨Φ(c), a⟩ := ⟨α_{c}, a_{C}(c)⟩,

for any a ∈ g. Here ⟨·, ·⟩ denotes the natural pairing between a vector space and its dual and a_{C} denotes the vector field on C corresponding to the infinitesimal action of g, a_{C}(c) = \frac{d}{dt}_{|t=0}(exp(ta).c). Φ is easily seen to be equivariant with respect to the given action on C and the coadjoint action on g^{∗}. Moreover Φ satisfies the so-called moment condition

dΦ_{a} = −i_{a_{C}}dα.

In the case of a contact manifold (C, H) we consider the natural bundle homomorphism α: T C → L = T C/H. We consider α as an L-valued differential 1-form and define the corresponding moment map as a section in the bundle g^{∗}⊗L over C. By abuse of notation we denote by g^{∗} here the trivial vector bundle C × g^{∗} over C. Notice that the action g.(c, λ) := (g.c, Ad_{g}^{∗}(λ)) gives g^{∗} the structure of a G-bundle over C. Since H ⊆ T C is G-invariant, L carries the structure of a G-bundle as well. So Y := g^{∗} ⊗ L is a G-bundle. The moment map Φ: C → Y = Hom(g, L) given by

Φ_{a}(c) = ⟨Φ(c), a⟩ = ⟨α_{c}, a_{C}(c)⟩ ∈ L_{c}

turns out to be equivariant and moreover satisfies a certain moment condition as well (see proposition 2).

The first aim of this note is to describe the natural reduction procedure for contact structures and exact contact structures in the case where the Lie group acts freely and properly. In analogy to the symplectic and exact symplectic case we call this the Marsden-Weinstein reduction.

Theorem 1 (Reduction for contact manifolds). Let (C, H) be a contact manifold and let G be a Lie group acting by contactomorphims freely and properly on C. Let L = T C/H, Φ: C → Y := g^{∗} ⊗ L the induced moment map and o ⊆ Y be the image of the zero section of Y → C. Then the following holds:

(a) M := Φ^{−1}(o) is a smooth invariant submanifold of C;

(b) i^{♯}H := i_{∗}^{−1}(i^{∗}H) ⊆ T M defines a distribution of constant rank on M, where i: M ↪ C denotes the inclusion map;

(c) there exists a unique distribution H_{0} ⊆ T C_{0} of hyperplanes with π^{♯}H_{0} = i^{♯}H, where π: M → C_{0} := M/G denotes the natural G-fibration. Moreover H_{0} is non-degenerate and defines therefore a contact structure on C_{0}.

Observe that in this contact case a natural moment map always exists. This is in contrast to the symplectic case where a moment map corresponding to a symplectic action may not exist (cf. [3] for the obstructions which could occur).

Theorem 2 (Reduction for exact contact manifolds). Let (C, α) be an exact contact manifold and let G be a Lie group acting by exact contactomorphisms freely and properly on C. Let Φ: C → g^{∗} be the natural moment map. Then the following holds:

(a) M := Φ^{−1}(0) is a smooth and invariant submanifold of C;

(b) there exists a unique 1-form α_{0} on C_{0} with π^{♯}α_{0} = i^{♯}α, where i: M ↪ C denotes the inclusion and π: M → C_{0} := M/G denotes the natural Gfibration. Moreover α_{0} defines an exact contact structure on C_{0}.

The corresponding statement in the exact symplectic case is also true. More precisely: If (M, α) is an exact symplectic manifold (i.e. ω := −dα is nondegenerate) where a Lie group G acts freely and properly by exact symplectomorphisms, then there exists a natural moment map and the symplectic reduction M_{0} carries a canonical exact symplectic structure α_{0} (see [4]).

As is well known (see [1], [2] e.g.), one associates to any contact manifold (C, H) its so-called (exact) symplectification (P, β) which is the R^{∗}-principal bundle π: P → C corresponding to the line bundle L^{∗} → C together with a canonical 1-form β. Here L^{∗} denotes the dual of the line bundle L. It turns out that an action of G by contactomorphisms on C lifts to an action on P by exact symplectomorphisms such that π is equivariant. Moreover the action of G stays free and proper. Finally the induced moment map Φ_{P}: P → g^{∗}, ⟨Φ_{P} , a⟩ = ⟨β, a_{P} ⟩ for p ∈ P, a ∈ g, is compatible with the moment map Φ_{C}: C → g^{∗} ⊗ L in the sense that

⟨Φ_{P} (p), a⟩ = p(⟨Φ_{C} ∘ π(p), a⟩),

for all a ∈ g and p ∈ P. Thus one may reduce (P, β) by the exact symplectic reduction procedure and obtains an exact symplectic manifold (P_{0}, β_{0}). On the other hand one can reduce (C, H) by contact reduction to (C_{0}, H_{0}) and it is worthwile to mention that (P_{0}, β_{0}) is the symplectification of (C_{0}, H_{0}). Thus symplectification and reduction commutes. (However, this is not the way how theorem 1 is proved. In fact, it is technically and conceptually simpler to stay in the contact category itself.)

The second aim of the paper is to investigate the case where G acts properly but not necessarily freely. So for a point c ∈ C the isotropy group K = G_{c} = {g ∈ G | g.c = c} may be non-trivial. (However, it is compact.) In this situation one would like to prove a so-called slice theorem, i.e., one looks for a certain model space D together with an embedding j: G/K ↪ D such that G-invariant neighborhoods U of j(G/K) ⊆ D and V of i(G/K) ⊆ C, i: G/K ↪ C, gK ↦ g.c, are Gequivariantly contactomorphic.

As in the symplectic case (cf. [7] e.g.) such a normal form will be given for points in the zero moment level M = Φ^{−1}(o) ⊆ C, where Φ is the natural moment map. Suppose that c ∈ M. Then the G-orbit of c is necessarily tangential, i.e., gc := {a_{C}(c) | a ∈ g} is contained in H_{c} ⊆ T C_{c}. In fact, by the definition of α and Φ one sees immediately that 0 = Φ_{a}(c) = ⟨α, a_{C}⟩(c), i.e., a_{C}(c) ∈ ker α_{c} = H_{c}. Moreover gc ⊆ H_{c} is isotropic with respect to the I -valued symplectic form ω_{c}: H_{c} × H_{c} → I. Here I denotes the quotient space T C_{c}/H_{c} which is a K - module in a natural way. Of course, ω_{c} is K -equivariant. One considers now the perpendicular gc^{⊥} of gc ⊆ H_{c} with respect to ω_{c} and defines

E := gc^{⊥}/gc.

Then the triple (I, E, τ), where τ: E × E → I is the non-degenerate skewsymmetric and K -equivariant bilinear form induced from ω_{c}, is called a contact slice for G at c. Associated to (I, E, τ) we construct a canonical contact structure on the vector bundle

D := G ×_{K} (k^{0} ⊗ I × E × I).

Here k^{0} ⊆ g^{∗} denotes the annihilator of k ⊆ g and D is the G-bundle over G/K associated to the K -module k^{0} ⊗I ×E ×I via the diagonal K -action with respect to the principal K -bundle G → G/K. The induced moment map turns out to be the map Φ: D → g^{∗} ⊗ I ≅ (k^{0} ⊕ k^{∗}) ⊗ I,

Φ([g, (λ, v, w)]) = Ad_{g}^{∗}(λ + Φ_{E}(v)).

Here G acts on g^{∗} ⊗ I = Hom(g, I) by the coadjoint action in the first factor and Φ_{E}: E → I is the natural moment map according to the I -valued symplectic structure on E, i.e., the homogeneous quadratic polynomial

⟨Φ_{E}(v), b⟩ = \frac{1}{2}τ(b.v, v)

for b ∈ k.

Theorem 3 (Local contact slice theorem). Let (C, H) be a contact manifold and let G act properly on C by contactomorphisms. Let Φ be the induced moment map and c ∈ C in the zero moment level of Φ. Denote by (I, E, τ) the contact slice of G in c. Then there exists G-invariant neighborhoods U of the zero section j: G/K ↪ D and V of i: G/K ↪ C, gK ↦ g.c, and a G-equivariant contactomorphism f: U → V with f ∘ j = i.

In the special case where I is the trivial K -module (e.g., if K is connected this must be the case since K is compact and I is 1-dimensional) the contact slice (E, τ) gives also rise to the space P = G ×_{K} (k^{0} × E), which serves as a model space for the symplectic case. In fact, P carries even the structure of an exact symplectic manifold and G acts by exact symplectomorphisms. Now for any exact symplectic manifold (P, β) it is well-known (see e.g. [1], [2]) that there exists a natural (exact) contactification of P, namely C = P ×R with the contact form α = dz − π_{1}^{∗}β (where π_{1} is the projection onto the first factor and z is a coordinate on R). As it turns out the model space D for the contact case is just the contactification of the model space P for the symplectic case.

2. Reduction of contact manifolds and exact contact manifolds

Let (C, α) be an exact contact manifold and let a Lie group G act by exact contactomorphisms, g^{♯}α = α for all g ∈ G. The natural moment map is given by the map Φ: C → g^{∗}, Φ_{a} = ⟨Φ, a⟩ for a ∈ g and

Φ_{a} = ⟨α, a_{C}⟩,

where a_{C} denotes the vector field on C corresponding to a ∈ g. As in the exact symplectic case it is now seen that Φ is equivariant with respect to the given action on C and the coadjoint action on g^{∗}. In fact, since the exponential map exp: g → G is equivariant with respect to the adjoint action Ad on g and the conjugation action on G, one finds that

(Ad_{g}(a))_{C}(g.c) = \frac{d}{dt}_{|t=0}(g exp(ta)g^{−1}.g.c) = g_{∗}(a_{C})(c)

and therefore

⟨Ad_{g^{−1}}^{∗}(Φ(g.c)), a⟩ = ⟨Φ(g.c), Ad_{g}(a)⟩ = ⟨α_{g.c}, (Ad_{g}(a))_{C}(g.c)⟩
= ⟨α_{g.c}, g_{∗}(a_{C})(c)⟩ = ⟨g^{♯}α_{c}, a_{C}(c)⟩ = ⟨α, a_{C}⟩(c) = Φ_{a}(c),

thus Φ(g.c) = Ad_{g}^{∗}(Φ(c)).

Similarly the moment condition for Φ transfers to the exact contact case. Since g^{♯}α = α for all g ∈ G, the Lie derivative of α in the orbit direction vanishes, L_{a_{C}}α = 0 for all a ∈ g. So

0 = L_{a_{C}}α = i_{a_{C}}dα + di_{a_{C}}α.

Denoting ω := −dα this implies

dΦ_{a} = i_{a_{C}}ω

as in the symplectic case.

Proposition 1. Let (C, α) be an exact contact manifold as above and Φ: C → g^{∗} its associated moment map. For any c ∈ Φ^{−1}(0) the following is true:

(a) (i) If S_{c} := ker(DΦ_{c}), then S_{c} is transversal to H_{c} = ker(α_{c}), S_{c} + H_{c} = T C_{c};

(ii) if K_{c} := S_{c} ∩ H_{c}, then the infinitesimal G-orbit gc is contained in K_{c}

and is the kernel of the restriction of ω_{c} := −dα_{c} to K_{c}×K_{c}, K_{c}^{⊥} = gc;

(b)

ran(DΦ_{c}) = k^{0},

where k^{0} ⊆ g^{∗} is the annihilator of the infinitesimal isotropy k = {a ∈ g | a_{C}(c) = 0} of c and ran(DΦ_{c}) denotes the image of DΦ_{c}.

Proof. (a) Recall that by definition of the contact structure the restriction of dα_{c} to H_{c} × H_{c} is non-degenerate, since

dα(ξ, η) = −α([X, Y]),

if X, Y are extensions of ξ, η with values in H. Therefore the kernel of dα: T C_{c} × T C_{c} → R, R_{c} := ker(dα_{c}), is 1-dimensional and transversal to H_{c}, H_{c} ⊕ R_{c} = T C_{c}. By the moment condition R_{c} is obviously contained in ker DΦ_{c} = S_{c}; thus S_{c} is transversal to H_{c}.

Now let ξ ∈ K_{c} = S_{c} ∩ H_{c}. Then 0 = ⟨DΦ_{c}, ξ⟩, which means 0 = dα(a_{C}(c), ξ) for all a ∈ g. Thus ξ ∈ K_{c} if and only if ξ ∈ gc^{⊥} ⊆ H_{c}. On the other hand the orbit Gc is in M = Φ^{−1}(0) by the equivariance of Φ, thus gc ⊆ S_{c}; and gc ⊆ H_{c} by the very definition of Φ, which shows that gc ⊆ gc^{⊥} = K_{c}. The kernel of the restriction of dα_{c} to K_{c} × K_{c} is therefore gc^{⊥⊥} = gc.

(b) It is clear that ran(DΦ_{c}) ⊆ k^{0} since for any a ∈ k by definition a_{C}(c) = 0 and therefore

⟨DΦ_{c}(ξ), a⟩ = ⟨dΦ_{a}(c), ξ⟩ = −dα(a_{C}(c), ξ) = 0

for all ξ ∈ T C_{c}. On the other hand dim ker(DΦ_{c}) = (2n−dim(g/k)+1 by part (a) and therefore dim ranDΦ_{c} = dim g/k = dim k^{0} which shows that ran(DΦ_{c}) = k^{0}.

Corollary. (a) If G acts freely, then 0 ∈ g^{∗} is a regular value of Φ and M = Φ^{−1}(0) is a dim G-codimensional and invariant submanifold of C;

(b) If G acts freely and properly, then C_{0} := M/G is a manifold and π: M → C_{0} is a G-principal bundle.

Proof. (a) The equivariance of Φ shows the invariance of M ⊆ C and, by part (b) of the proposition, Φ is submersive. So the result follows from the implicit function theorem.

(b) If G acts freely and properly on C, it acts freely and properly on M as well. Thus M → M/G is a G-principal bundle (by the standard slice theorem for proper actions).

Consider now a contact manifold (C, H) and a Lie group G acting on C by contact diffeomorphisms, i.e., g^{♯}H = H. Thus H ⊆ T C is a G-subbundle and the quotient bundle L = T C/H inherits the structure of a G-bundle as well. The natural L-valued 1-form, i.e., the bundle homomorphism α: T C → L given by the canonical projection, is then G-equivariant. Now the associated moment map, i.e., the section Φ: C → g^{∗} ⊗ L given by Φ_{a} = ⟨Φ, a⟩,

Φ_{a} = ⟨α, a_{C}⟩,

is equivariant with respect to the given action on C and the diagonal action on the tensor bundle Y = g^{∗} ⊗ L. In particular the inverse image of the zero section o ⊆ Y ,

M = Φ^{−1}(0),

is G-invariant since o ⊆ Y is G-invariant.

Fix now c_{0} ∈ M and suppose that G acts properly on M. Then the local slice theorem (see [6]) guarantees that there exists an invariant neighborhood U of c_{0} where the line bundle L is G-equivariantly trivial, i.e., there exists a Gequivariant and nowhere vanishing section s: U → L|U. This implies that for every a ∈ g there exists a unique function Ψ_{a}: U → R such that

Φ_{a}(c) = Ψ_{a}(c)s(c)

for all c ∈ U. Similarly there exists a unique and G-invariant 1-form β on U, β ∈ E^{1}(U), so that

α|U = β · s.

Now one sees that Ψ_{a} = ⟨β, a_{C}⟩ and, as in the exact case, a computation shows that

dΨ_{a} = −i_{a_{C}}dβ.

To put this formula into a global form recall that for any L-valued 1-form α: T C → L one can define the exterior differential dα as an L-valued skewsymmetric bilinear form on H = ker(α), i.e., dα: H × H → L by the following (cf. [5]). If U ⊆ C is an open set where there exists a nowhere vanishing section s of L, then α = βs for a unique 1-form β ∈ E^{1}(U). Another such section s̃ is of the form s̃ = φ^{−1}s for φ ∈ E^{∗}(U) and then α = β̃s̃ with β̃ = φβ. But dβ̃ = dφ ∧ β + φ dβ; so dβ̃|H = φ dβ|H since H = ker(β), and thus

dβ̃ · s̃ = dβ · s,

showing that dα (= dβ · s on U) is well defined. A contact structure on a manifold C can then be seen as a contact structure (H, ω) on the tangent bundle. i.e., H ⊆ T C is a hyperplane bundle and

ω: H × H → L := T C/H

is a non-degenerate skew-symmetric bilinear form, satisfying the closeness condition

−dα = ω,

where α: T C → L is the canonical projection.

But now observe that in our case, where α: T C → L is a G-equivariant bundle homomorphism not only the contraction of ω = −dα by an element X with values in H is well defined as a bundle morphism on H, i.e., i_{X}ω: H → L, but moreover, if X comes from the group action on C, i.e., X = a_{C} with a ∈ g, the contraction i_{a_{C}}(−dα) makes sense even on the whole tangent bundle T C along M, i.e., ia_{C}(−dα)_{c}: T C_{c} → L_{c}. In fact, suppose that U ⊆ C is G-invariant and L → C trivializes G-equivariantly over U as above via an equivariant section s. Then α = βs for an invariant 1-form β on U and one sets

i_{a_{C}}(−dα) := i_{a_{C}}(−dβ) · s.

Namely, if s = φs̃ with another equivariant nowhere vanishing section s̃, then necessarily φ is G-invariant. Infinitesimally this means ⟨dφ, a_{C}⟩ = 0 for all a ∈ g. Moreover, if c ∈ M = Φ^{−1}(o), then ⟨β, a_{C}⟩(c) = 0 by the definition of the moment map and therefore for α = β̃s̃ with β̃ = φβ we see that

i_{a_{C}}dβ̃ = i_{a_{C}}(dφ ∧ β + φ dβ)
= ⟨dφ, a_{C}⟩β − ⟨β, a_{C}⟩dφ + φ i_{a_{C}}dβ = φ i_{a_{C}}dβ,

showing that the definition of i_{a_{C}}(−dα)_{c}: T C_{c} → L_{c} is well-behaved for c ∈ M.

Finally, as for any vector bundle, the tangent space of L in an element o_{c} ∈ L_{c} of the zero section splits canonically into

T L_{o_{c}} = T C_{c} ⊕ L_{c},

and we let π_{2}: T L_{o_{c}} → L_{c} the projection onto the second factor. Now, using that the moment map Φ: C → Y writes Φ|U = Ψs, where Ψ is the moment map for the exact contact manifold (U, β), one sees that the moment condition in the case of a contact transformation group reads as follows.

Proposition 2. If G acts on a contact manifold (C, H) by contact diffeomorphisms and if Φ is the associated moment map, then for any c ∈ M = Φ^{−1}(o) we have the moment condition

π_{2} ∘ (dΦ_{a})_{c} = −(i_{a_{C}}dα)_{c}.

The reader will have no difficulty to prove the statements analogous to those given in proposition 1 for the non-exact contact case.

Proposition 3. Let (C, H) be a contact manifold and Φ: C → g^{∗} ⊗ L (with L = T C/H) the associated moment map. Let o ⊆ Y = g^{∗} ⊗ L be the zero section and c ∈ M = Φ^{−1}(o). Then the following is true:

(a) (i) If S_{c} = DΦ_{c}^{−1}(T o_{Φ(c)}) ⊆ T C_{c}, then S_{c} is transversal to H_{c}, H_{c} + S_{c} =

T C_{c}; (ii) if K_{c} := S_{c} ∩ H_{c}, then the infinitesimal G-orbit gc in contained in K_{c} and in fact the kernel of the restriction of ω_{c} = −dα_{c}: H_{c} × H_{c} → L_{c} to

K_{c} × K_{c} → L_{c} ≅ S_{c}/K_{c}.

(b) If k ⊆ g is the isotropy algebra of c, then

ran(π_{2} ∘ DΦ_{c}) = k^{0},

the annihilator of k in g^{∗}.

Corollary. (a) If G acts freely on C, then Φ: C → Y = g^{∗} ⊗ L is transversal to the zero section o ⊆ Y and M = Φ^{−1}(0) is a dim G-codimensional and invariant submanifold of C.

(b) If G acts freely and properly, then C_{0} = M/G is a manifold and π: M → C_{0} is a G-principal bundle. At this point we want to recall what we mean by a contact vector space

(cf. [5]). It is given by a triple (V, H, ω), where V is a (real) vector space, H ⊆ V is a hyperplane and ω: H ×H → V/H is a non-degenerate skew-symmetric bilinear map. The situation we arrived at above is the following. If W ⊆ V is a transversal subspace, i.e., W + H = V , then (W, K, τ) with K := W ∩ H and τ = ω|K × K → V/H ≅ W/K is what we call a precontact space, i.e., K is a hyperplane and τ is skew-symmetric (not necessarily non-degenerate). One may reduce a precontact space (W, K, τ) in a natural way by the following. Denote by N ⊆ K the kernel of τ and further W_{0} := W/N, K_{0} := K/N, and τ_{0}: K_{0} × K_{0} → W_{0}/K_{0} ≅ W/K the induced structure from τ. It is clear that (W_{0}, K_{0}, τ_{0}) is a contact vector space.

We are now able to prove theorem 1 and theorem 2 in the introduction.

Proof. Let (C, H) be a (possibly non-exact) contact manifold and consider the L-valued 1-form α: T C → L := T C/H. Suppose that the Lie group G acts by contact transformations freely and properly and let Φ: C → g^{∗} ⊗ L, Φ_{a} = ⟨α, a_{C}⟩, be the induced moment map. We have already seen that the preimage of the zero section o ⊆ g^{∗} ⊗ L, i.e., M = Φ^{−1}(0), is G-invariant and in fact a G-principal bundle over the quotient C_{0} = M/G by the natural projection π: M → C_{0}.

Now fix c_{0} ∈ C_{0} and choose an element c ∈ M in the fibre of π over c_{0}. The contact structure on T C_{c}, which is denoted by (H_{c}, ω_{c}) as usual, induces a precontact structure on T M_{c} = S_{c} as was shown above, K_{c} = H_{c} ∩ S_{c}, τ_{c} = ω_{c}|K_{c}×K_{c}. Moreover, by proposition 3, the kernel of τ_{c} is exactly the infinitesimal G-orbit gc ⊆ K_{c}. Therefore, by the preceding remark, T M_{c}/gc carries the structure (K_{0}, τ_{0}) of a contact vector space in a natural way. On the other hand the differential Dπ_{c}: T M_{c} → (T C_{0})_{c_{0}} induces an isomorphism

f_{c}: T M/gc → (T C_{0})_{c_{0}}.

Thus we define a contact structure ((H_{0})_{c_{0}}, (ω_{0})_{c}) on (T C_{0})_{c_{0}} such that f_{c} is in fact a contact isomorphism.

Next we observe that this construction of (H_{0}, ω)_{c_{0}} is independent of the choice of the preimage c ∈ π^{−1}(c_{0}). Indeed, if c̃ is another preimage, then c = g.c̃ for a group element g ∈ G and by the invariance of the original contact structure H on C one finds that the construction of (H_{0}, ω_{0})_{c_{0}} on (T C_{0})_{c_{0}} is in fact well defined.

Finally we have to check that (H_{0}, ω_{0}) does in fact define a contact structure on C_{0} (not only on the tangent bundle T C_{0}). So let α_{0}: T C_{0} → L_{0} := T C_{0}/H_{0} be the induced L_{0}-valued 1-form. We have to check that −dα_{0}: H_{0} × H_{0} → L_{0} satisfies −dα_{0} = ω_{0}, which is the closeness condition for contact manifolds. But π^{♯}(ω_{0}) = i^{♯}(ω) by construction of the structure ω_{0}, where i: M ↪ C is the inclusion, and π^{♯}α_{0} = i^{♯}α by definition of H_{0} ⊆ T C_{0}. Therefore

π^{♯}(−dα_{0}) = d(−π^{♯}α_{0}) = −d(i^{♯}α) = −i^{♯}(dα) = i^{♯}ω = π^{♯}ω_{0},

and by the fact that π: M → C_{0} is a submersion we conclude that −dα_{0} = ω_{0}. We have proved theorem 1.

Proof. Let (C, α) be an exact contact manifold, G a Lie group acting by exact contactomorphisms freely and properly and Φ: C → g^{∗}, Φ_{a} = ⟨α, a_{C}⟩, be the induced moment map. The zero level M = Φ^{−1}(0) is a G-invariant submanifold and a G-principal bundle over the quotient C_{0} = M/G by proposition 1, π: M → C_{0}. Moreover, the pullback i^{♯}α on M, i: M ↪ C denoting the inclusion, defines an exact precontact manifold in the sense that K = ker(i^{♯}α) ⊆ T M is a hyperplane bundle but τ := −d(i^{♯}α) : K×K → R is not necessarily non-degenerate. However, in our case it is of constant rank and its kernel is given exactly by the direction of the G-orbits, i.e.,

ker(d(i^{♯}α)_{c}) = gc,

again by proposition 1. For any c_{0} ∈ C_{0} and c ∈ π^{−1}(c_{0}) we denote by Dπ_{c}: T M_{c} → (T C_{0})_{c_{0}} the differential of π in c as before. Thus there exists a unique homomorphism (α_{0})_{c_{0}}: (T C_{0})_{c_{0}} → R with (α_{0})_{c_{0}} ∘ Dπ_{c} = α_{c}. Moreover, there exists a unique skew-symmetric bilinear map (ω_{0})_{c_{0}}: (T C_{0})_{c_{0}} × (T C_{0})_{c_{0}} → R with (ω_{0})_{c_{0}} ∘ (Dπ_{c} × Dπ_{c}) = τ_{c} by the reduction procedure for precontact spaces.

We have to check that the definition of (α_{0})_{c_{0}} and (ω_{0})_{c_{0}} on (T C_{0})_{c_{0}} is independent of the chosen preimage c ∈ π^{−1}(c_{0}). This follows from the invariance of α and ω and is left to the reader. We have defined now the global 1-form α_{0} and the global 2-form ω_{0} on C_{0}. Finally we have to check that ω_{0} = −dα_{0}. But again π^{♯}α_{0} = i^{♯}α and π^{♯}(ω_{0}) = τ = i^{♯}(ω) = i^{♯}(−dα) by construction and therefore

π^{♯}(−dα_{0}) = −d(π^{♯}α_{0}) = −d(i^{♯}α) = i^{♯}(−dα) = π^{♯}(ω_{0}),

showing that −dα_{0} = ω_{0} and proving theorem 2.

Example. If M is a manifold (of dimension n + 1), m ∈ M, then any hyperplane c ⊆ T M_{m} is called a contact element on M. The space of all contact elements on M is therefore the projectivized (P^{n}-) bundle C = P(T^{∗}M). It carries a canonical (non-exact) contact structure given by: ξ ∈ H_{c} ⊆ T C_{c} if π_{∗}ξ ∈ c, where π: C → M is the bundle projection. Let G be a Lie group acting on M by smooth diffeomorphisms. Then G acts on P(T^{∗}M)) by bundle homomorphisms as well. Denote by E_{m} = gm ⊆ T M_{m} an infinitesimal orbit. It is seen directly from the definition of the moment map on C that c ∈ π^{−1}(m) is in the zero moment level if and only if c ⊇ E_{m}. Thus Φ^{−1}(o) = P(E^{0}), where E^{0} ⊆ T^{∗}M denotes the annihilator of E. It is clear that E_{m}^{0} ≅ (T M_{m}/E_{m})^{∗} and therefore isomorphic to (T M_{0})_{m_{0}}^{∗}, when G is acting freely and properly on M and M_{0} is its orbit space M/G (m_{0} = Gm). This shows that the contact reduction of the manifold of all contact elements on M is nothing else than the manifold of all contact elements on the reduction M_{0},

(P(T^{∗}M))_{0} = P(T^{∗}M_{0}).

Example. A partial differential equation of the first order on an open set Ω ⊆ R^{n} is given by an equation F (x, u, p) = 0 on Ω×R×R^{n} and a solution is a smooth function u: Ω → R so that F (x, u(x), Du(x)) = 0 for all x ∈ Ω. A generalization of this concept to a manifold M of dimension n is given by a hypersurface Γ in the space J^{1}M of 1-jets of functions on M. A 1-jet of a smooth function f on M at p ∈ M is by definition the element j^{1}f_{p} := f mod m_{p}^{2} ∈ E_{p}/m_{p}^{2}, where E_{p} denotes the germs of smooth functions at p and m_{p} ⊆ E_{p} its maximal ideal. J^{1}M comes along with a natural vector bundle structure π: J^{1}M → M of rank n + 1, j^{1}f_{p} ↦ p. Now C := J^{1}M carries a natural exact contact structure given by the 1-form α, α_{c}(ξ) = π_{∗}ξ(c) − pr_{∗}(ξ), where pr : J^{1}M → R is given by pr(j^{1}f_{p}) = f(p). (π_{∗}ξ ∈ T M_{p} acts as a derivation on the 1-jet c = j^{1}f_{p} ∈ π^{−1}(p).) For every function f on M one has the associated section j^{1}f: M → J^{1}M =: C, p ↦ j^{1}f_{p}, which defines a Legendre submanifold of C, i.e., an integral submanifold of the contact distribution of C of maximal dimension n. Given a hypersurface Γ ⊆ C we say that f is a solution if the image of its section j^{1}f is contained in Γ.

Now if a Lie group G acts by (exact) contact transformations (freely and properly) on C and if Γ is a G-invariant hypersurface, it is clear how the reduction of C reduces the problem of finding a solution of the partial differential equation given by Γ to an appropriate Γ_{0} on the reduced space C_{0}. In particular, if the action is induced from a G-action on M itself, it is not hard to see that C_{0} coincides with the 1-jet bundle of M_{0} = M/G.

3. A local slice theorem for contact transformation groups

We consider now a contact manifold (C, H), on which a Lie group G acts properly by contactomorphisms but not necessarily freely. Thus the isotropy groups G_{c} = {g ∈ G | g.c = c} ⊆ G are not necessarily trivial. The aim of this section is to find a normal form of the G-action in the neighborhood of an orbit lying in the zero moment level M = Φ^{−1}(o) ⊆ C.

The basic ingredient is a tubular neighborhood theorem for contact geometry as it is proved in [5]. For convenience we repeat here its equivariant version. Suppose that G acts on (C, H) properly by contactomorphisms as above and let B be a G-manifold and i: B ↪ C an equivariant tangential embedding, meaning that i_{∗}T B ⊆ i^{∗}H. One observes then that i_{∗}T B is an isotropic subbundle of i^{∗}H with respect to the i^{∗}L-valued structure i^{∗}ω: i^{∗}H × i^{∗}H → i^{∗}L where ω = −dα, α: T C → L = T C/H. In fact, this is just the fact that the Lie bracket of two vector fields along i with values in i_{∗}T B (i.e., vector fields on B) is again a vector field along i with values in i_{∗}T B. Therefore one can build the so-called symplectic normal bundle E := i_{∗}T B^{⊥}/i_{∗}T B over B. In fact, it is a G-vector bundle together with an I -valued and equivariant symplectic structure τ: E ×E → I := i^{∗}L inherited from i^{∗}ω. The tangential embedding theorem asserts that the triple (I, E, τ) characterizes a neighborhood of i(B) ⊆ C up to contactomorphisms. More precisely, we say that two triples (I_{1}, E_{1}, τ_{1}) and (I_{2}, E_{2}, τ_{2}) are isomorphic, if there exist equivariant bundle isomorphisms θ: I_{1} → I_{2} and η: E_{1} → E_{2} such that τ_{2} ∘ (η × η) = θ ∘ τ_{1}.

Theorem (Equivariant tangential embedding). Suppose that i_{1}: B ↪ (C_{1}, H_{1}) and i_{2}: B ↪ (C_{2}, H_{2}) are two equivariant tangential embeddings with isomorphic symplectic normal bundles (I_{1}, E_{1}, τ_{1}) and (I_{2}, E_{2}, τ_{2}). Then there exist invariant neighborhoods U_{1} ⊆ C_{1} of i_{1}(B) and U_{2} ⊆ C_{2} of i_{2}(B) and an equivariant contactomorphism f: U_{1} → U_{2} satisfying f ∘ i_{1} = i_{2}.

For the proof one has to check that the various steps in the proof of the tangential embedding theorem carry over to the equivariant case which could be done.

So what we have to do in order to determine the neighborhood geometry of a G-orbit in Φ^{−1}(o) ⊆ C is to identify its symplectic normal bundle. In fact, by the very definition of Φ: C → g^{∗} ⊗ L, a G-orbit in M = Φ^{−1}(o) is tangential. Moreover, Gc ≅ G/K (Gc denoting the orbit {g.c | g ∈ G} and K the isotropy in c, K = G_{c}), and the symplectic normal bundle (I, E, τ̄) over Gc is a K -bundle. As such it is of the form

I = G ×_{K} I, E = G ×_{K} E

for the K -vector spaces I and E sitting as the fibres over c. Thus, all we have to do is to look for the triple (I, E, τ), where I = T C_{c}/H_{c}, E = gc^{⊥}/gc and τ: E × E → I is the symplectic structure induced from i^{∗}ω_{c}: H_{c} × H_{c} → L_{c}. We have called (I, E, τ) the contact slice G at c.

The rest of this section is therefore devoted to the construction of a certain model space D associated to the datum (I, E, τ), which will be a contact manifold where G acts by contactomorphisms and an equivariant tangential embedding j: G/K ↪ D such that the contact slice at K ∈ G/K is exactly (I, E, τ).

So let I be a 1-dimensional K -module, let E be a 2r-dimensional K - module and let τ: E ×E → I be a non-degenerate skew-symmetric and equivariant bilinear form. As a first step we consider the K -line bundle I over G associated to I, i.e., I := G × I together with the action

k.(g, w) = (gk^{−1}, k.w).

On the other hand right multiplication on G induces a K -structure on the tangent bundle T^{∗}G → G and thus we can form the tensor product T^{∗}G⊗I as a K -bundle via the diagonal action. The fibre over an element g ∈ G is identified in a natural way with Hom(T G_{g}, I),

(T^{∗}G ⊗ I)_{g} = Hom(T G_{g}, I);

thus T^{∗}G ⊗ I may be seen as the I -valued covectors on G. If one identifies T^{∗}G via left multiplication on G with G × g^{∗}, i.e.,

(g, λ) ↦ (DL_{g})_{1}^{∗}(λ)

(where L_{g}: G → G denotes left multiplication with g on G), then the action of K on T^{∗}G ⊗ I ≅ G × g^{∗} ⊗ I = G ⊗ Hom(g, I) reads

k.(g, λ) = (gk^{−1}, k.λ),

where k.λ(a) = k.λ(Ad_{k^{−1}}(a)) for a ∈ g.

The next step is to see that T^{∗}G ⊗ I carries a canonical I -valued 1-form β. In terms of the trivialization G × Hom(g, I) the tangent space in a point (g, λ) may be identified with T G_{g} × Hom(g, I) ≅ g × Hom(g, I) (again identified by left multiplication). The form β is then given simply by

β_{g,λ}(a, μ) = ⟨λ, a⟩.

It is worthwile to note that just as a vector bundle over G (forgetting the K -action for a moment) T^{∗}G ⊗ I is isomorphic to T^{∗}G since I = R and also the 1-form β is the canonical one. However, the K -bundle T^{∗}G ⊗ I is different from T^{∗}G (if the action of K on I is not the trivial one). The essential point is then that the canonical I -valued 1-form is K -equivariant with respect to the action T^{∗}G ⊗ I and the action on I.

Now consider the whole datum (I, E, τ). We choose a basis (ξ_{1}, …, ξ_{r}, η_{1}, …, η_{r}) of E and (ζ) of I with corresponding coordinates (x^{1}, …, x^{r}, y^{1}, …, y^{r}) = (x, y) and z such that the form τ: E × E → I reads

τ = dx ∧ dy · ζ.

Then one sees immediately that the I -valued 1-form

β := \frac{1}{2}(y dx − x dy) · ζ

is not only a potential of τ, i.e., −dβ = τ, but moreover, β is K -equivariant as well. Furthermore β is independent of the chosen Darboux-coordinates.

For the next step consider the product P = (T^{∗}G ⊗ I) × E with its projectors π_{1}, π_{2} onto the two factors. Denote by β_{1} the canonical I -valued 1-form on T^{∗}G ⊗ I and β_{2} the I -valued 1-form on E just constructed. The sum β := π_{1}^{♯}β_{1} + π_{2}^{♯}β_{2} gives an I -valued 1-form on P. The product

C = P × I

with its projection pr_{1} onto the first factor gets now the structure of a contact manifold via the I -valued 1-form

α = dz · ζ − pr_{1}^{♯}β.

By construction α is K -equivariant and defines a moment map Φ_{R}: C → k^{∗} ⊗I = Hom(k, I). Via the trivialization C = G × Hom(g, I) × E × I it reads

Φ_{R}(g, λ, v, w) = −λ|k + Φ_{E}(v).

Here Φ_{E}: E → k^{∗} ⊗ I = Hom(k, I) is the natural moment map of E with respect to β, i.e., for b ∈ k and Φ_{b} = ⟨Φ_{R}, b⟩ it is

Φ_{b}(v) = ⟨β_{v}, b.v⟩ = \frac{1}{2}τ(b.v, v),

which is homogeneous quadratic in v. A choice of a K -invariant scalar product on g (recall that K is compact since the action is proper) allows us to embed k^{∗} as a K -submodule of g^{∗} via the orthogonal complement of k^{0} ⊆ g^{∗}, g^{∗} = k^{0} ⊕k^{∗}. Therefore we may identify Φ_{R}^{−1}(0) with G × (k^{0} ⊗ I × E × I) via

(g, λ, v, w) ↦ (g, λ + Φ_{E}(v), v, w).

The contact quotient of C by K, i.e.,

D := G ×_{K} (k^{0} ⊗ I × E × I),

inherits a canonical I -valued 1-form α_{0}. Observe that the bundle G ×_{K} I, i.e., the cocontact bundle of the contact structure H_{0} = ker(α_{0}) is now not trivial any longer (if I is not trivial, of course). This is the construction of the local model together with its embedding j: G/K ↪ D, which is the zero section of the vector bundle D.

Finally one has to look for the group action of G coming from left multiplication on G. Since it commutes with right multiplication, the whole group action goes through the construction. The G-moment map Φ: D → g^{∗}⊗I = (k^{0}⊕k^{∗})⊗I reads

Φ([g, λ, v, w]) = g.(λ + Φ_{E}(v)).

That is the local normal form of the moment map. Up to the coadjoint action on Hom(g, I) it is linear in the first summand and quadratic in the second (“linear” and “angular” momentum).

In fact, in order to prove theorem 3 from section 1 it is now only necessary to convince oneself that the contact slice of j(K) ∈ D, j: G/K ↪ D, is exactly (I, E, τ), which follws from the construction. An application of the equivariant tangential embedding theorem yields the result.

References

[1] Arnol’d, V. I., “Mathematical Methods of Classical Mechanics,” Springer-Verlag, Second Edition, Berlin Heidelberg New York, 1989.

[2] Arnol’d, V. I., and A. B. Givental’, Symplectic geometry, in: “Dynamical systems IV” (Eds.: V. I. Arnol’d, S. P. Novikov), Springer-Verlag, Berlin Heidelberg New York, 1980.

[3] Guillemin, V. and S. Sternberg, “Symplectic techniques in physics,” Cambridge University Press, reprinted with corrections, Cambridge, 1990.

[4] Loose, F., Liouville forms in a neighborhood of an isotropic embedding, Ann. Inst. Fourier 47 (1997), 257–272.

[5]] Loose, F., The tubular neighborhood theorem in contact geometry, Abh. Math. Sem. Univ. Hamburg 68 (1998), 129–147.

[6] Palais, R., On the existence of slices for actions of non-compact Lie groups, Ann. Math. 73 (1961), 295–323.

[7] Sjamaar, R. and E. Lerman, Stratified symplectic spaces and reduction, Ann. Math. 134 (1991), 375–422.

[8] Weinstein, A., “Lectures on symplectic manifolds,” Regional conf. series in math. 29, AMS, Rhode Island, 1977.

Frank Loose Mathematisches Institut der Universität Auf der Morgenstelle 10 D-72076 Tübingen frank.loose@uni-tuebingen.de

Received May 25, 1998 and in final form August 31, 2000