Vector fields and differential forms on the orbit space of a proper action

In this paper we study differential forms and vector fields on the orbit space of a proper action of a Lie group on a smooth manifold, defining them as multilinear maps on the generators of infinitesimal diffeomorphisms, respectively. This yields an intrinsic view of vector fields and differential forms on the orbit space.


Introduction
This paper is a complete version of [1] which includes the correction to Lemma 3.1, whose proof contained an error, which was pointed out by Prof. G. Schwarz. This paper is part of a series of papers devoted to the study of the geometry of singular spaces in terms of the theory of differential spaces, which were introduced by Sikorski [2], see also [3]. In this theory, geometric information about a space S is encoded in a ring C ∞ (S) of real valued functions, which are deemed to be smooth. In particular, we are concerned with the class of subcartesian spaces introduced by Aronszajn [4]. A Hausdorff differential space S is subcartesian if every point x of S has a neighborhood U that is diffeomorphic to a subset V of a Euclidean (Cartesian) space R n . The restriction of C ∞ (S) to U is isomorphic to the restriction of C ∞ (R n ) to V , see [5].
Palais [6] introduced the notion of a slice for an action of a not necessarily compact Lie group G on a manifold M . Since then, the structure of the space M/G of orbits of a proper action of G on M has been investigated by many mathematicians. In [7] Duistermaat showed that M/G is a subcartesian differential space with differential structure C ∞ (M/G) consisting of push forwards of smooth G invariant functions on M by the G orbit mapping π : M → M/G.
On a smooth manifold M , there are two equivalent definitions of a vector field, namely, as a derivation of C ∞ (M ), or as a generator of a local one parameter local group of diffeomorphisms of M . Choosing one, and proving the other is a matter of preference. On a subcartesian differential space S, which is not in general a manifold, these notions differ. We use the term vector field on S for a generator of a local one parameter group of local diffeomorphisms and denote the class of all vector fields on S by X(S). A key reason for the choice made in this paper is the special case of the orbit space of a proper action. The class of derivations of C ∞ (S) is, in general, larger than the class X(S). For S = M/G we show that a derivation Y of C ∞ (M/G) is in X(S), that is, is a vector field on M/G, if and only if there exists a G invariant vector field X on M such that Y is π related to X, that is, Y • π = T π • X, where π : M → M/G is the G orbit map.
In the literature there has been extensive discussion about the notion of a differential form on a singular space, see Smith [8], Marshall [9], and Sjamaar [10]. Here, in our search for an intrinsic notion of a differential form, we have been led to see them as multilinear maps on vector fields. In the case of a 1-form θ on M/G with θ a linear mapping θ : X(M/G) → C ∞ (M/G) : Y → θ|Y over the ring C ∞ (M/G) of smooth functions on M/G, which is to say θ|f Y = f θ|Y for every f ∈ C ∞ (M/G). With this definition we show that every differential 1-form on M/G pulls back under the orbit map π to a semi-basic G invariant 1-form on M . Furthermore every G invariant semi-basic differential 1-form on M is the pull back by π of a differential 1-form on M/G. We define a differential exterior algebra of differential forms on the orbit space, which satisfies a version of de Rham's theorem. Our version is larger than Smith's as it includes forms that are not Smith forms, see section 6. It also handles singular orbit spaces of a proper action of a Lie group on a smooth manifold. The Lie group need not be compact and the orbit space need not be smooth, both of which Koszul hypothesized in [11].
We now give a section by section description of the contents of this paper. Section 2 deals with basic properties of a proper action of a Lie group G on a smooth manifold M and the differential structure of the orbit space M/G. We introduce the reader to the theory of subcartesian differential spaces in the context of the orbit space M/G. The differential geometry of M is described in terms of its smooth structure given by the ring C ∞ (M ) of smooth functions on M . The differential geometry of the orbit space M/G, which may have singularities, is similarly described in terms of the ring C ∞ (M/G) of smooth functions on M/G, which is isomorphic to the ring C ∞ (M ) G of smooth, Ginvariant functions on M . Since a proper action has an invariant Riemannian metric, several results are proved using properties of the geodesics of the metric. Also certain objects are shown to be smooth submanifolds.
In section 3 we study vector fields on the orbit space M/G. In the case of the manifold M , derivations of the ring C ∞ (M ) are vector fields on M , and they generate a local one parameter group of local diffeomorphisms of M . In the case of the ring C ∞ (M/G), not all derivations of C ∞ (M/G) generate local one parameter groups of local diffeomorphisms of M/G. The derivations of C ∞ (M/G), which generate local one parameter groups of local diffeomorphisms of M/G. This is the key idea of this paper. We establish that every vector field on the quotient M/G is covered by a G-invariant vector field on M . It is well known that the space M/G is stratified, see [7,12,13]. We show that every vector field on M/G defines a vector field tangent to each stratum of M/G.
In section 4, we define differential 1-forms on the orbit space M/G as linear mappings on the the space X(M/G) of smooth vector fields on M/G. The most important consequence of this definition relates to pulling back 1-forms from M/G to M . In particular, our notion of a differential 1-form is intrinsic.
In section 5 to prove a version of de Rham's theorem we enlarge the algebra of differential 1-forms to k-forms with an exterior derivative operator. The key techinical point is that everything is developed in terms of the Lie derivative of vector fields. Almost all of this section looks the same as that on manifolds.
In section 6 we give all the details of the simplest nontrivial example. This example reveals that differential forms in our sense are not the same as those of Smith [8].

Basic properties
This section gives some of the basic properties of smooth vector fields on the orbit space of a proper action of a Lie group on a smooth manifold.
Let M be a connected smooth manifold with a proper action of a Lie group G on M , and let be the orbit map of the G action Φ.
Let C ∞ (M ) G be the algebra of smooth G invariant functions on M and let is a bijective algebra isomorphism, whose inverse is Proposition 1. The orbit space M/G with the differential structure C ∞ (M/G) is a locally closed subcartesian differential space.
Let X be a smooth vector field on a manifold M . X gives rise to a map called the derivation associated to X. If we want to emphasize this action of vector fields on M , we say that they form the space Der C ∞ (M ) of derivations of C ∞ (M ). If we want to emphasize that X generates a local one parameter group of local diffeomorphisms of M , we say that X is a vector field on M and write X(M ) for the set of vector fields on M . For each smooth manifold M we have X(M ) = Der C ∞ (M ). However, these notions need not coincide for a subcartesian differential space.
Let (S, C ∞ (S)) be a differential space with X a derivation of C ∞ (S). Let be a maximal integral curve of X, which starts at x. Here I x is an interval containing 0. If t, s, and t + s lie in I x , and if s ∈ I ϕ X t (x) and t ∈ I ϕ X s (x) , then The map ϕ X t may fail to be a local diffeomorphism of the differential space S, see example 3.2.7 in [5, p.37]. A vector field on a subcartesian differential space S is a derivation X of C ∞ (S) such that for every x ∈ S there is an open neighborhood U of x and ε > 0 such that for every t ∈ (−ε, ε) the map ϕ X t is defined on U and its restriction to U is a diffeomorphism from U onto an open subset of S. In other words, the derivation X is a vector field on S if t → ϕ X t is a local one parameter group of local diffeomorphisms of S. Example 1. Consider Q ⊆ R with the structure of a differential subspace of R. Let ι : Q → R be the inclusion mapping. The differential structure C ∞ (Q) of Q consists of ι * f , which is the restriction of a smooth function f on R to Q. Let X(x 1 ) = a 1 (x 1 ) ∂ ∂x1 be a vector field on R. Then for every f ∈ C ∞ (R) and every . We now show that we can obtain ι * ∂f ∂x1 by operations on Q 2 . Let x 0 1 ∈ Q and let {(x 1 ) n } be a sequence of points in Q, which converges to x 0 1 . Then Thus we have shown that ι * (X(f )) = X |Q (ι * f ) for every f ∈ C ∞ (R). In other words, the restriction X |Q of the vector field X to Q is a derivation of C ∞ (Q). Thus Der C ∞ (Q) = {X |Q X ∈ X(R)}. However, no two distinct points of Q can be joined by a smooth curve. Hence only the derivation of C ∞ (Q) that is identically 0 on Q admits integral curves, that is, Let X(M ) G be the set of smooth G invariant vector fields on M , that is, be the local flow of X, that is, ϕ is a differentiable mapping such that Here D is a domain, that is, D is the largest (in the sense of containment) open This leads to the module homomorphism Since Y is a linear mapping of C ∞ (M/G) into itself, it follows that it is a derivation of C ∞ (M/G). We now show that the map X → π * • X • π * is a module homomorphism.
The importance of the module homomorphism (5) stems from the following result.
Proof. Because the orbit space M/G is locally closed and subcartesian, every maximal integral curve of X projects under the G orbit map to a maximal integral curve of Y . It follows that Y is a smooth vector field on M/G, see proposition 3.2.6 on page 34 of [5].
The following example shows that not every derivation on C ∞ (M/G) is a vector field on M/G.
is not a smooth vector field on R/Z 2 , because its maximal integral curve γ σ0 starting at σ 0 ∈ R/Z 2 , given by γ σ0 (t) = t + σ 0 , is defined on [−σ 0 , ∞), which is not an open interval that contains 0.
Fix m ∈ M . Then G m = {g ∈ G Φ g (m) = m} is the isotropy group of the action Φ at m. It is a compact subgroup of G, see Duistermaat and Kolk [14]. Let H be a compact subgroup of G. The set is a submanifold of M , which is not necessarily connected. Hence its connected component are submanifolds. Connected components of M H are H invariant submanifolds of M , see Duistermaat and Kolk [14]. The conjugacy class in G of a closed subgroup H is denoted by (H) = {gHg −1 ∈ G g ∈ G} and is called a type. The set is called an orbit type (H  Proof. See page 118 of Duistermaat and Kolk [14]. Proof. Let m ∈ M (H) . Then G m = gHg −1 for some g ∈ G. Let ϕ t be the local 1 parameter group of local diffeomorphisms of M generated by X ∈ X(M ) G . Then Let Y be a smooth vector field on M , which is π related to the smooth G invariant vector field X on M , that is, Note that the orbit type M (H) need not be connected and its connected components may be of different dimensions. In the following we concentrate our attention on the properties of the connected components of M (H) , which we denote by M (H) . Proof. See page 74 of [5].  We need the next few results to prove this theorem, which is the main result of this section.
Proof. Let S m be a slice to the G action Φ at m. By Bochner's lemma, see [15, p.306], there is a local diffeomorphism ψ : T m M → M , which sends 0 m ∈ T m M to m ∈ M and intertwines the H = G m action

using the restriction of the G invariant Riemannian metric on
M to ker T m π |M (H) , see Palais [6]. Because the vector space (ker T m π |M (H) ) ⊥ is isomorphic to T m M (H) , having the same dimension, it follows that the map T m π |M (H) is surjective. Consequently, the map π |M (H) is a submersion.
Next we construct a connection on the fibration π |M ( where ver m = ker T m π |M ( where m = π(m), is an isomorphism of vector spaces. Thus Equations (9) and (10) define an Ehresmann connection E on the fibration π |M (H) : and T m Φ g ver m ⊆ ver g·m imply ver g·m = T m Φ g ver m and hor g·m = T Φ g hor m , the distributions ver and hor are G invariant. Thus the connection E is G invariant. Let which gives rise to the Hamiltonian system (E, T * M (H) , ω), where ω is the canonical symplectic form on T * M (H) . The Hamiltonian vector field X E on be the local flow of the vector field X E , which is defined for t in an open interval There is an open tubular neighborhood U of the zero section of the cotangent bundle ρ : , see Brickell and Clark [16].
Next we reduce the G symmetry of the Hamiltonian system (E, , ω). The reduced system has a Hamiltonian vector field X E defined by Consider the mappings and the projection mapping Then τ V m is a local trivialization of the fibration defined by the mapping π 1 because for every n ∈ (π |M (H) ) −1 (m) and every We now show that τ V m is a diffeomorphism. Define the smooth maps The following calculation shows that Clearly, τ V is a diffeomorphism. It intertwines the G action Φ with the G action and Proof. To see this let ξ ∈ g, the Lie algebra of G.
So the vector field Y on U × G and the vector field Y U on U are π 1 related.
Thus the G invariant vector field X on V is π |V related to the vector field Y U on U .
Proof of theorem 1. We just have to piece the local bits together. Cover the orbit type Thus the G invariant vector fields X Vi piece together to give a smooth G invariant vector field X on M (H) . Since X Vi is π |Vi related to the vector field Y Ui , the vector field X on M (H) is π |M (H) related to the vector field Y on M (H) .

Vector fields on M/G
We start with a local argument in a neighbourhood of a point m ∈ M with compact isotropy group H. By Bochner's lemma, there is a local diffeomorphism ψ : T m M → M, which sends 0 m ∈ T m M to m ∈ M and intertwines the linear H action Because the action Φ on the smooth manifold M is proper, it has a G invariant Riemannian metric k. Using the restriction of k to T m M , we define be a set of generators for the algebra of H invariant polynomials on R n . Let y = (y 1 , ..., y ℓ ) be coordinates on R ℓ . The orbit map of of the H action on R n is ρ : R n → R n /H : x → y = (σ 1 (x), . . . , σ ℓ (x)).
for every x ∈ R n . Smooth functions on the orbit space R n /H are restrictions to R n /H of smooth functions on R l . For every f ∈ C ∞ (R n /H), the pull-back ρ * f by the orbit map is given by The H orbit map ρ : R n → R n /H ⊆ R ℓ is a smooth map of differential spaces. Consider an open ball B ⊆ R n , centered at the origin 0 ∈ R n , with closure B. We are interested in B/H, the space of H orbits in B. Restricting ρ to the domain B ⊆ R n and the codomain Σ = ρ(B) ⊆ R ℓ gives which is a surjective smooth map of differential spaces.
Proof. Let Y be a smooth vector field on B/H, which extends to a derivation on Σ = B/H. Since B/H is a closed differential subspace of R ℓ , in coordinates y = (y 1 , ..., y ℓ ) on R ℓ , we may write is a locally trivial fibration, whose fibre over Repeating the above argument at each point y ∈ B (K) /H leads to a covering {W α } α∈I of B (K) by H invariant open subsets W α of B (K) on which there exists an H invariant vector field X Wα , which is ρ related to the restriction of the vector field Y to V α = W α /H. Using an H invariant partition of unity on B (K) , we obtain a vector field X B (K) on B (K) , which is ρ |B (K) related to Y |B (K) /H , i.e., The module X(B) H of H invariant smooth vector fields on B is finitely generated by polynomial vector fields, see [17], and we denote a generating set by {X j } N j=1 . Hence, every H invariant smooth vector field X B on B is of the form X B = N j=1 h j X j for some h 1 , . . . , h n ∈ C ∞ (B). Similarly, every K invariant smooth vector field on B (K) can be written as need not extend to a smooth function on B. Therefore a generic vector field on B (K) need not extend to a smooth vector field on B. On the other hand, the H invariant vector field X B (K) on B (K) , is obtained above from a smooth bounded vector field Y on B/H. Therefore for each x ∈ B (K) ⊆ B ⊆ R n , where every k j ∈ C ∞ (R n ), and k j|B (K) is the restriction of k j to B (K) .
Since B (K) is open and dense in B, we may define provided that lim k→∞ k j (x k ) exists and is unique. Since the vector fields X j (x k ) are smooth on B, ∂t (c(t))dt.
Thus, the values of k j on B are uniquely determined by k j|B (K) . Repeating this argument for all the first order partial derivatives of k j , we deduce that the first order partial derivatives of k j on B are uniquely determined by k j|B (K) and its first partial derivatives. Continuing this process for every partial derivative of every order shows that the restriction of k j to B is uniquely determined by k j|B (K) .
The above argument applies to each of the functions k i , for i = 1, . . . , n, in Equation (19)  x k ∈ B (K) for all k ∈ Z ≥1 . Then y ∈ ρ B (x) ∈ B (J) /H and because Y |B (K)/H is the restriction to B (K) /H of a smooth, and hence continuous, vector field on B/H. By Proposition 7, for every orbit type B (J) , the manifold B (J) /H is an invariant manifold of of the vector field Y . So Y |B (J) /H is a vector field on B (J)/H . Hence, for every x ∈ B, Therefore, X is a smooth vector field on B, which is ρ B related to the vector field Y on B/H.  Because S m is a slice, G · U m = {Φ g (U m ) ∈ M g ∈ G} is a G invariant open subset of M , which contains the G orbit G · m. On G · U m define the vector field X = {(Φ g ) * X g ∈ G}. We check that X is well defined. Suppose that g · s m = g ′ · s ′ m , where g, g ′ ∈ G and s m , s ′ m ∈ S m . Since S m is a slice, it follows that g −1 g ′ = h ∈ H. Hence

The aim of the rest of this section is to prove
where the last equality above follows because the vector field X is H invariant. So the vector field X on G · U m is well defined and by definition is G invariant.
Next we show that X is smooth. Let L be a complement to the Lie algebra h of the Lie group H in the Lie algebra g of the Lie group G. For every ξ ∈ L and η ∈ h consider the map µ : G → (exp L)H : exp(ξ + η) → exp ξ exp η, which sends the identity element e G of G to e G · e H = e G . It is a local diffeomorphism, since its tangent T eG µ : g → L ⊕ h = g is the identity map. Thus there are open subsets V G , V L , and V H of e G , 0 L , and e H , respectively, such that µ(V G ) = exp V L · V H . Hence every g ∈ V G may be written uniquely as g = (exp ξ)h for some ξ ∈ V L and some h ∈ V H . For every s m ∈ U m ⊆ S m we have Consider the local diffeomorphism For s ∈ S m let e t Xm (s) be the integral curve of the vector field X m starting at s. Since X m is a G invariant extension of X m to a vector field on G · S m (whose smoothness we want to prove) of a smooth H invariant vector field X m on S m , it follows that X m |S m = X m is a smooth vector field on S m . Therefore for all s ∈ S m . Consider a curve c q in U ⊆ G·S m starting at q = Φ exp ξ (s) defined by c q (t) = Φ exp ξ ϕ Xm t (Φ −1 exp ξ (q)) . Using Equation (22) we obtainċ q (0) = T q Φ exp ξ ( X m (s)) = X(q) for all q ∈ U . Since the family of curves t → c q (t) depends smoothly on q ∈ U and U is an open subset of G · S m containing S m , it follows that X U is a smooth vector field on U . For any m ′ ∈ G · S m there exists a g ∈ G such that the open set Φ g (U ) contains m ′ . Since X is G invariant, smoothness of X on U ensures that X is smooth on Φ g (U ). Hence X is a smooth vector field on G · S m .
The above argument can be repeated at each point m ∈ M . This leads to a covering {G · S mα } α∈I of M by open G invariant subsets G · S mα , where S mα is a slice at m α for the action of G on M and I is an index set. If Y is a vector field on M , then for each α ∈ I there exists a G invariant vector field X mα on G · S mα that is π related to the restriction of Y to (G · S mα )/G ⊆ M/G. Using a G invariant partition of unity on M subordinate to the covering {G · S mα } α∈I , we can glue the pieces X mα together to obtain a smooth G invariant vector field X on M , which is π related to the vector field Y on M/G. Proposition 9 If Y is a derivation of C ∞ (M/G), which is π related to a derivation X of C ∞ (M ) G , then Y is a smooth vector field on M/G.

Proof.
Since M is a smooth manifold, X is a smooth G invariant vector field on M , which is π related to derivation Y of C ∞ (M/G). Thus the image under π of a maximal integral curve of X on M , is a maximal integral curve of Y on M/G. Hence Y is a smooth vector field on the locally closed subcartesian differential space M/G, C ∞ (M/G) .

Differential 1-forms on the orbit space
In this section we define the notion of a differential 1-form on the orbit space M/G of a proper group action Φ : G × M → M : (m, g) → g · m on a smooth manifold M with orbit map π : M → M/G : m → m = G · m. We show that the differential 1-forms on M/G together with the exterior derivative generate a differential exterior algebra.
Theorem 2 and Proposition 9 show that Y is a vector field on M/G if and only if there is a G invariant vector field X on M , which is π related to Y , that is, every integral curve of Y is the image under the map π of an integral curve of X. Let Λ 1 (M/G) be the set of differential 1-forms on M/G, that is, the set of linear mappings which are linear over the ring C ∞ (M/G), that is, θ|f Y = f θ|Y for every f ∈ C ∞ (M/G) and every Y ∈ X(M/G).
In order to prove some basic properties of differential 1-forms on M/G, we need to prove some properties of the G orbit map π (2).

The map
where X ∈ X(M ) G and Y is the vector field on M/G constructed in Proposition 2, is the tangent of the map π at m ∈ M . To show that T m π is well defined we argue as follows. Suppose that v m = X ′ (m), where X ′ ∈ X(M ) G . Then since T m π is a linear map.
where g is the Lie algebra of G.
Proof. By definition π −1 (m) = G · m. Thus The curve γ m : To prove the reverse inclusion, we argue as follows. ker T m π. There is a vector field X on M with X(m) = v m having an integral curve γ p : where the equality follows from Equation (24). This verifies Equation (23).
A differential 1-form ω on M is semi-basic with respect to the G action Φ if and only if X ξ ω = 0 for every ξ ∈ g, the Lie algebra of G.
Proposition 11. Let ϑ be a G invariant semi-basic differential 1-form on M . Then there is a 1-form θ on M/G such that ϑ = π * θ.
Proof. Given Y ∈ X(M/G), there is an X ∈ X(M ) G , which is π related to Y , that is, T m πX(m) = Y (π(m)) for every m ∈ M . It is clear that the definition of θ needs to be π * ( θ|Y ) = ϑ|X .
It remains to show that θ is well defined. Since the 1-form ϑ and the vector field X are G invariant, we get for every (g, m) ∈ G×M . Thus the function M → R : m → ϑ|X (m) is smooth and G invariant. We now show that the mapping θ : where θ is given in Equation (26), is well defined. Suppose that since the 1-form ϑ on M is semi-basic. This shows that the map θ : (26) it follows that θ is a linear mapping and that θ|f Y = f θ|Y for every f ∈ C ∞ (M/G). Hence θ is a differential 1-form on M/G, that is, θ ∈ Λ 1 (M/G). Every X ∈ X(M ) G is π related to a Y ∈ X(M/G). Thus π * θ|X = π * ( θ|Y ) = ϑ|X , that is, ϑ = π * θ.

de Rham's theorem
In this section we construct an exterior algebra of differential forms on the orbit space M/G with an exterior derivative d and show that de Rham's theorem holds for the sheaf of differential exterior algebras.
Proof. The proof is analogous to the proof of Proposition 10 for 1-forms on M/G and is omitted.
Proof. The proof is analogous to the proof of Proposition 11 for G invariant semi-basic 1-forms on M and is omitted.
We now define the exterior algebra Λ(M/G) of differential forms on M/G. Let θ ∈ Λ h (M/G) and φ ∈ Λ k (M/G). The exterior product is the h + k form θ ∧ φ on M/G corresponding to the G invariant semi-basic h + k-form π * θ ∧ π * φ on M . Then Λ(M/G) = ℓ ⊕Λ ℓ (M/G), ∧ is an exterior algebra of differential forms on M/G.

The exterior derivative operator d on Λ(M/G) is defined in terms of the Lie bracket of vector fields on
The following lemma shows that this Lie bracket is well defined. (27) Proof. We compute.
from which Equation (27) follows, because the orbit map π is surjective.
[ . ] is a Lie bracket on X(M/G).
Proof. The corollary follows from a computation using Equation (27). We give another argument. Bilinearity of the Lie bracket is straightforward to verify. We need only show that the Jacobi identity holds. We compute.
which is the Jacobi identity on X(M/G).
Let ϕ be an ℓ-form on M/G. Inductively define the exterior derivative d of ϕ as the (ℓ + 1)-form given by To complete the definition of exterior derivative, we define d on 0-forms. This we do as follows.
Lemma 12. Let G be a Lie group, which acts linearly on R n by Φ : G × R n → R n . Let H be a compact subgroup of G. Let β be an H invariant closed ℓform with ℓ ≥ 1 on an open H invariant ball B centered at the origin of R n , whose closure is compact. Suppose that β is semi-basic with respect to the G action Φ, that is, X ξ β = 0 for every ξ ∈ g, the Lie algebra of G. Here X ξ (x) = T e Φ m ξ. Then there is an H invariant (ℓ − 1)-form α on B, which is semi-basic with respect to the G action Φ, such that β = dα.
Proof. Let X be a linear vector field on R n all of whose eigenvalues are negative real numbers. By averaging over the compact group H, we may assume that X is H invariant. Let ϕ t be the flow of X, which maps B into itself. Moreover, The (ℓ − 1)-form α = ∞ 0 ϕ * t (X β) dt on B is H invariant, since ϕ t commutes with the H action on B, and X β is an H invariant (ℓ−1)-form on B, because the vector field X and the ℓ-form β are both H invariant. Thus β = dα on B. Moreover, α is G semi-basic, since The last equality above follows because the ℓ-form β is G semi-basic.
Since M/G is a locally contractible space, we have with the H ⊆ G action Φ on U m . Let ϑ be the semi-basic G invariant form on G · U m such that (π |G·Um ) * θ = ϑ. Since θ is closed by hypothesis, it follows that the semi-basic ℓ-form ϑ on G · U m is closed. Let φ = ϑ|U m . Then φ is a semi-basic H invariant closed ℓ-form on U m . Under the map ψ the ℓ-form φ pulls back to a G semi-basic for every s ∈ U m and every v s ∈ T s S m . Arguing as in the proof of Lemma 6, it follows that σ is a smooth G invariant (ℓ − 1)-form on G · U m . The form σ is semi-basic. Moreover, dδ = ϑ on G · U m , since for every g ∈ G one has Let U m = π(U m ). Since U m is contractible and the G orbit map π is continuous and open, it follows that the open neighborhood U m of m ∈ M/G is contractible.
Since the ℓ-form σ is semi-basic, there is an ℓ-form φ on U m such that π * φ = σ on G · m. On G · U m we have Because the orbit map π is surjective, it follows that θ = dφ on U m , which proves the proposition. To prove de Rham's theorem, we will need some sheaf theory, which can be found in appendix C of Lukina, Takens, and Broer [18]. Let U = {U α } α∈I be an open covering of M/G. Because M/G is locally contractible, the open covering U has a good refinement U ′ , that is, every U β ∈ U ′ with β ∈ J ⊆ I is locally contractible and U β1 ∩ · · · ∩ U βn is either contractible or empty. In addition, because M/G is paracompact, every open covering has a locally finite subcovering. Since the G action on M is proper, the orbit space M/G has a C ∞ (M/G) partition of unity subbordinate to the covering U.
Define the differential exterior algebra valued sheaf Λ over M/G by Λ : U α → Λ(U α ), ∧, d |Uα , whose sections are differential forms on U α . The sheaf Λ induces the subsheaves whose sections are differential ℓ-forms on U α . Note that Let R be the sheaf of locally constant R-valued functions on M/G. The two exact sequence of sheaves 0 → R → Λ → · · · and 0 → R → Λ ℓ → · · · are exact.
We say that the sheaf Λ is fine if for every open subset U of M/G, every smooth function f on M/G and every smooth section s : Theorem 3. The sheaves Λ and Λ ℓ of sections of the vector bundles Λ and Λ ℓ are fine.
Proof. We treat the case of the sheaf Λ. The proof for the sheaf Λ ℓ is similar and is omitted. The definition of fineness holds by definition of differential form. We are now in position to formulate de Rham's theorem. Let Λ ℓ be the sheaf of differential ℓ-forms on M/G and let d : Λ ℓ → Λ ℓ+1 be the sheaf homomorphism induced by exterior differentiation. For each ℓ ∈ Z ≥0 let Z ℓ = ker d, whose elements are closed ℓ-forms on M/G. By Lemma 13 Z 0 = R. Define the ℓ th Proof. We give a sketch, leaving out the homological algebra, which is standard. For more details, see [18] or [19]. Let U be a good covering of M/G. The Consider the Z 2 action on R 2 generated by The algebra of Z 2 invariant polynomials on R 2 is generated by the polynomials σ 1 (x) = x 2 1 , σ 2 (x) = x 2 2 , and σ 3 (x) = x 1 x 2 , which are subject to the relation Let be the Hilbert map of the Z 2 action associated to the polynomial generators σ 1 (x), σ 2 (x), and σ 3 (x). The map σ (36) is the orbit map of the Z 2 action on R 2 . The relation defines the orbit space R 2 /Z 2 as a closed semialgebraic subset Σ of R 3 with coordinates (σ 1 , σ 2 , σ 3 ). Geometrically Σ is a cone in R 3 with vertex (0, 0, 0).
By Theorem 2 every smooth vector field on Σ is σ related to a smooth Z 2 invariant vector field on R 2 . Because the C ∞ (R 2 ) Z2 module X(R 2 ) Z2 of smooth Z 2 invariant vector fields on R 2 is generated by the vector fields X i for 1 ≤ i ≤ 4 given by Equation (39), it follows that the σ related vector fields Y i for 1 ≤ i ≤ 4 given by Equation (40) generate the C ∞ (Σ) module X(Σ) of smooth vector fields on Σ.
Let θ 4 be the 1-form on Σ defined by its values Here σ i = σ i|Σ for i = 1, 2, 3. The 1-form θ 4 is not the restriction of a 1-form on R 3 to Σ.
Proof. Equation (45) follows immediately from the definition of θ i given in Equation (44).
We give three proofs of the assertion about θ 4 .