ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 10, Pages 1–10 (February 18, 2004) S 1079-6762(04)00124-6

COMPLETING LIE ALGEBRA ACTIONS TO LIE GROUP ACTIONS

FRANZ W. KAMBER AND PETER W. MICHOR

(Communicated by Alexandre Kirillov)

Abstract. For a finite-dimensional Lie algebra 𝔤 of vector fields on a manifold M we show that M can be completed to a G-space in a universal way, which however is neither Hausdorff nor T_{1} in general. Here G is a connected Lie group with Lie-algebra 𝔤. For a transitive 𝔤-action the completion is of the form G/H for a Lie subgroup H which need not be closed. In general the completion can be constructed by completing each 𝔤-orbit.

1. Introduction. In [7], Palais investigated when one could extend a local Lie group action to a global one. He did this in the realm of non-Hausdorff manifolds, since he showed that completing a vector field X on a Hausdorff manifold M may already lead to a non-Hausdorff manifold on which the additive group ℝ acts. We reproved this result in [3], being unaware of Palais’ result. In [4] this result was extended to infinite dimensions and applied to partial differential equations like Burgers’ equation: solutions of the PDE were continued beyond the shocks and the universal completion was identified.

Here we give a detailed description of the universal completion of a Hausdorff 𝔤-manifold to a G-manifold. For a homogeneous 𝔤-manifold (where the finitedimensional Lie algebra 𝔤 acts infinitesimally transitive) we show that the Gcompletion (for a Lie group G with Lie algebra 𝔤) is a homogeneous space G/H for a possibly non-closed Lie subgroup H (Theorem 7). In Example 8 we show that each such situation can indeed be realized. For general 𝔤-manifolds we show that one can complete each 𝔤-orbit separately and replace the 𝔤-orbits in M by the resulting G-orbits to obtain the universal completion _{G}M (Theorem 9). All 𝔤invariant structures on M ‘extend’ to G-invariant structures on _{G}M. The relation between our results and those of Palais are described in Section 10.

2. 𝔤-manifolds. Let 𝔤 be a Lie algebra. A 𝔤-manifold is a (finite-dimensional Hausdorff) connected manifold M together with a homomorphism of Lie algebras ζ = ζ^{M} : 𝔤 → 𝔛(M) into the Lie algebra of vector fields on M. We may assume without loss of generality that it is injective; if not, replace 𝔤 by 𝔤/ ker(ζ). We shall also say that 𝔤 acts on M.

\begin{matrix}c \\ ◯ \end{matrix}2004 American Mathematical Society

Received by the editors October 27, 2003.

2000 Mathematics Subject Classification. Primary 22F05, 37C10, 54H15, 57R30, 57S05.

Key words and phrases. 𝔤-manifold, G-manifold, foliation.

FWK and PWM were supported by ‘Fonds zur Förderung der wissenschaftlichen Forschung, Projekt P 14195 MAT’.

The image of ζ spans an integrable distribution on M, which need not be of constant rank. So through each point of M there is a unique maximal leaf of that distribution; we also call it the 𝔤-orbit through that point. It is an initial submanifold of M in the sense that a mapping from a manifold into the orbit is smooth if and only if it is smooth into M; see [5, 2.14ff.].

Let ℓ : G × M → M be a left action of a Lie group with Lie algebra 𝔤. Let ℓ_{a} : M → M and ℓ^{x} : G → M be given by ℓ_{a}(x) = ℓ^{x}(a) = ℓ(a, x) = a.x for a ∈ G and x ∈ M. For X ∈ 𝔤 the fundamental vector field ζ_{X} = ζ_{X}^{M} ∈ 𝔛(M) is given by ζ_{X}(x) = −T_{e}(ℓ^{x}).X = −T_{(e,x)}ℓ.(X, 0_{x}) = −∂_{t}|_{0} exp(tX).x. The minus sign is necessary so that ζ : 𝔤 → 𝔛(M) becomes a Lie algebra homomorphism. For a right action the fundamental vector field mapping without minus would be a Lie algebra homomorphism. Since left actions are more common, we stick to them.

3. The graph of the pseudogroup. Let M be a 𝔤-manifold, effective and connected, so that the action ζ = ζ^{M} : 𝔤 → 𝔛(M) is injective. Recall from [1, 2.3] that the pseudogroup Γ(𝔤) consists of all diffeomorphisms of the form

Fl_{t_{n}}^{ζ_{Xn}} ∘ … ∘ Fl_{t_{2}}^{ζ_{X_{2}}} ∘ Fl_{t_{1}}^{ζ_{X_{1}}} |U,

where X_{i} ∈ 𝔤, t_{i} ∈ ℝ, and U ⊂ M are such that Fl_{t_{1}}^{ζ_{X_{1}}} is defined on U, Fl_{t_{2}}^{ζ_{X_{2}}} is defined on Fl_{t_{1}}^{ζ_{X_{1}}} (U), and so on.

Now we choose a connected Lie group G with Lie algebra 𝔤, and we consider the integrable distribution of constant rank d = dim(𝔤) on G × M which is given by

(3.1)  {(L_{X}(g), ζ_{X}^{M}(x)) : (g, x) ∈ G × M, X ∈ 𝔤} ⊂ T G × T M,

where L_{X} is the left invariant vector field on G generated by X ∈ 𝔤. This gives rise to the foliation F_{ζ} on G × M, which we call the graph foliation of the 𝔤-manifold M.

Consider the following diagram, where L(e, x) is the leaf through (e, x) in G×M, O_{𝔤}(x) is the 𝔤-orbit through x in M, and W_{x} ⊂ G is the image of the leaf L(e, x) in G. Note that pr_{1} : L(e, x) → W_{x} is a local diffeomorphism for the smooth structure of L(e, x).

(3.2)  \xymatrix{ & L(e, x) \ar[dd] \ar[dr] \ar@{->>}[rr]^{^{pr^{2}}} &  & O_{𝔤}(x) \ar@{^{(}->}[d] \\  &  & G × M \ar[d]^{_{pr_{1}}} \ar[r]^{_{pr_{2}}} & M \\ [0, 1] \ar@{-->}[uur]^{c̃} \ar[r]^{c} & W_{x} \ar@{^{(}->}[r]^{open} & G}

Moreover we consider a piecewise smooth curve c : [0, 1] → W_{x} with c(0) = e and we assume that it is liftable to a smooth curve c̃: [0, 1] → L(e, x) with c̃(0) = (e, x). Its endpoint c̃(1) ∈ L(e, x) does not depend on small (i.e. liftable to L(e, x)) homotopies of c which respect the ends. This lifting depends smoothly on the choice of the initial point x and gives rise to a local diffeomorphism γ_{x}(c) : U → {e} × U → {c(1)} × U′ → U′, a typical element of the pseudogroup Γ(𝔤) which is defined near x. See [1, 2.3] for more information and Example 4 below. Note that the leaf L(g, x) through (g, x) is given by

(3.3)  L(g, x) = {(gh, y) : (h, y) ∈ L(e, x)} = (μ_{g} × Id)(L(e, x)),

where μ : G × G → G is the multiplication and μ_{g}(h) = gh = μ^{h}(g).

4. Examples. It is helpful to keep the following examples in mind, which elaborate upon [1, 5.3]. Let G = 𝔤 = ℝ^{2}, let W be an annulus in ℝ^{2} containing 0, and let M_{1} be a simply connected piece of finite or infinite length of the universal cover of W. Then the Lie algebra 𝔤 = ℝ^{2} acts on M but not the group. Let p : M_{1} → W be the restriction of the covering map, a local diffeomorphism.

Here G×_{𝔤} M_{1} ≅ G = ℝ^{2}. Namely, the graph distribution is then also transversal to the fiber of pr_{2} : G × M_{1} → M_{1} (since the action is transitive and free on M_{1}), thus describes a principal G-connection on the bundle pr_{2} : G × M_{1} → M_{1}. Each leaf is a covering of M_{1} and hence diffeomorphic to M_{1} since M_{1} is simply

connected. For g ∈ ℝ^{2} consider j_{g} : M_{1} ⟶^{ins_{g}} {g} × M_{1} ⊂ G × M_{1} ⟶^{π} G ×_{𝔤} M_{1}

and two points x ≠ y ∈ M_{1}. We may choose a smooth curve γ in M_{1} from x to y, lift it into the leaf L(g, x) and project it to a curve c in g + W from g to c(1) = g + p(y) − p(x) ∈ g + W. Then (g, x) and (c(1), y) are on the same leaf. So j_{g}(x) = j_{g}(y) if and only if p(x) = p(y). So we see that j_{g}(x) = g + p(x), and thus G ×_{𝔤} M_{1} = ℝ^{2}. This will also follow from 7.

Let us further complicate the situation by now omitting a small disk in M_{1} so that it becomes non-simply connected but still projects onto W, and let M_{2} be a simply connected component of the universal cover of M_{1} with the disk omitted. What happens now is that homotopic curves which act equally on M_{1} act differently on M_{2}.

It is easy to see with the methods described below that the completion_{G}M_{i} = ℝ^{2} in both cases.

5. Enlarging to group actions. In the situation of Section 3 let us denote by GM = G ×_{𝔤} M = G × M/F_{ζ} the space of leaves of the foliation F_{ζ} on G × M, with the quotient topology. For each g ∈ G we consider the mapping

(5.1)  j_{g} : M ⟶^{ins_{g}} {g} × M ⊂ G × M ⟶^{π} _{G}M = G ×_{𝔤} M.

Note that the submanifolds {g}×M ⊂ G×M are transversal to the graph foliation F_{ζ}. The leaf space_{G}M of G×M admits a unique smooth structure, possibly singular and non-Hausdorff, such that a mapping f : _{G}M → N into a smooth manifold N is smooth if and only if the compositions f ∘ j_{g} : M → N are smooth. For example we may use the structure of a Frölicher space or smooth space induced by the mappings j_{g} in the sense of [6, Section 23] on _{G}M = G ×_{𝔤} M. The canonical open maps j_{g} : M → _{G}M for g ∈ G are called the charts of _{G}M. By construction, for each x ∈ M and for g′g^{−1} near enough to e in G there exists a curve c : [0, 1] → W_{x} with c(0) = e and c(1) = g′g^{−1} and an open neighborhood U of x in M such that for the smooth transformation γ_{x}(c) in the pseudogroup Γ(𝔤) we have

(5.2)  j_{g′}|U = j_{g} ∘ γ_{x}(c).

Thus the mappings j_{g} may serve as a replacement for charts in the description of the smooth structure on _{G}M. Note that the mappings j_{g} are not injective in general. Even if g = g′, there might be liftable smooth loops c in W_{x} such that (5.2) holds. Note also some similarity of the system of ‘charts’ j_{g} with the notion of an orbifold where one uses finite groups instead of pseudogroup transformations.

The leaf space _{G}M = G×_{𝔤} M is a smooth G-space where the G-action is induced by (g′, x) ↦ (gg′, x) in G × M.

Theorem. The G-completion _{G}M has the following universal properties:

(5.3) Given any Hausdorff G-manifold N and 𝔤-equivariant mapping f : M → N there exists a unique G-equivariant continuous mapping f̃: _{G}M → N with f̃∘ j_{e} = f. Namely, the mapping f̄: G × M → N given by f̄(g, x) = g.f(x) is smooth and factors to f̃: _{G}M → N.

(5.4) In the setting of (5.3), the universal property holds also for the T_{1}-quotient of _{G}M, which is given as the quotient G×M/\overline{F}_{ζ} of G×M by the equivalence relation generated by the closure of leaves.

(5.5) If M carries a symplectic or Poisson structure or a Riemannian metric such that the 𝔤-action preserves this structure or is even a Hamiltonian action, then the structure ‘can be extended to _{G}M so that the enlarged G-action preserves these structures or is even Hamiltonian’.

Proof. (5.3) Consider the mapping f̄ = ℓ^{N} ∘ (Id_{G} ×f) : G × M → N which is given by f̄(g, x) = g.f(x). Then by (3.1) and (3.2) we have for X ∈ G

T f̄.(L_{X}(g), ζ_{X}^{M}(x)) = T ℓ.(L_{X}(g), T_{x}f.ζ_{X}^{M}(x))
= T ℓ.(R_{Ad(g)X}(g), 0_{f(x)}) + T ℓ(0_{g}, ζ_{X}^{N}(f(x)))
= −ζ_{Ad(g)X}(g.f(x)) + T ℓ_{g}.ζ_{X}^{N}(f(x)) = 0.

Thus f̄ is constant on the leaves of the graph foliation on G × M and thus factors to f̃ : _{G}M → N. Since f̄(g.g_{1}, x) = g.g_{1}.f(x) = g.f̄(g, x), the mapping f̃ is Gequivariant. Since N is Hausdorff, f̃ is even constant on the closure of each leaf, thus (5.4) holds also.

(5.5) Let us treat Poisson structure P on M. For symplectic structures or Riemannian metrics the argument is similar and simpler. Since the Lie derivative along fundamental vector fields of P vanishes, the pseudogroup transformation γ_{x}(c) in (5.2) preserves P. Since _{G}M is the quotient of the disjoint union of all spaces {g} × M for g ∈ G under the equivalence relation described by (5.2), P ‘passes down to this quotient’. Note that we refrain from putting too much meaning on this statement. □

The universal property (5.3) holds also for smooth G-spaces N which need not be Hausdorff, nor T_{1}, but should have tangent spaces and foliations so that it is meaningful to talk about 𝔤-equivariant mappings. We will not go into this, but see [6, Section 23], for some concepts which point in this direction.

As an application of the universal property of the G-completion _{G}M, we see that GM depends on the choice of G in the following way. We write G = Γ\G̃, where G̃ is the simply connected Lie group with Lie algebra 𝔤 and Γ ⊂ G̃ is the discrete central subgroup such that Γ ≅ π_{1}(G). Then we have _{G}M ≅ Γ\G̃M as G-spaces, so that G̃M is potentially less singular than _{G}M.

6. Example. Let 𝔤 = ℝ^{2} with basis X, Y, let M = ℝ^{3} \ {(0, 0, z) : z ∈ ℝ}, and let ζ^{α} : 𝔤 → 𝔛(M) be given by

(6.1)  ζ_{X}^{α} = ∂_{x} + α \frac{yz}{x^{2} + y^{2}} ∂_{z}, ζ_{Y}^{α} = ∂_{y} − α \frac{xz}{x^{2} + y^{2}} ∂_{z},  α > 0,

which satisfy [ζ_{X}^{α}, ζ_{Y}^{α}] = 0. By construction of the graph foliation F_{ζ^{α}} in (3.1) and the procedure summarized in diagram (3.2), the leaves of F_{ζ^{α}} are determined explicitly as follows. For any smooth curve c(t) = (ξ(t), η(t)) ∈ G starting at (ξ_{0}, η_{0}) we have ċ(t) = ξ̇(t) X + η̇(t) Y ∈ 𝔤 and the lifted curve (c(t), y(t)) is in the leaf L((ξ_{0}, η_{0}), y_{0}) if and only if it satisfies the first order ODE

(6.2)  (y(t), ẏ(t)) = ξ̇(t) ζ_{X}^{α}(y(t)) + η̇(t) ζ_{Y}^{α}(y(t))

with initial value y(0) = y_{0} = (x_{0}, y_{0}, u = z_{0}) ∈ M. Substituting (6.1) into (6.2), we see that this ODE is linear, that is, ẋ = ξ̇, ẏ = η̇ and ż = −α z \frac{xη̇−yξ̇}{r^{2}} = −α z \frac{xẏ−yẋ}{r^{2}}, where r^{2} = x^{2} + y^{2}. Thus the projection x(t) of y(t) to the (x, y)-plane is given by x(t) = c(t) − ((ξ_{0}, η_{0}) − x_{0}) = c(t) − (ξ_{0} − x_{0}, η_{0} − y_{0}), whereas the third equation leads to

(6.3)  z(t) = u e^{−α ∫_{0}^{t} dθ} = u e^{−α(θ(t)−θ_{0})} = ue^{αθ_{0}} e^{−αθ(t)} ,

where θ is the angle function in the (x, y)-plane. This depends only on the endpoints x_{0} , x(t) and the winding number of the curve x and is otherwise independent of x. Incompleteness occurs whenever the curve x goes to (0, 0) ∈ ℝ^{2} in finite time t̄ < ∞, that is, x(t) → (0, 0) , t ↑ t̄, or equivalently c(t) → (ξ_{0}, η_{0}) − x_{0} , t ↑ t̄. It follows that the leaf L((ξ_{0}, η_{0}), y_{0}) is parametrized by (r, θ) ∈ ℝ_{+} ×ℝ with z = z(θ) being independent of r > 0 and that

(6.4)  pr_{1} : L((ξ_{0}, η_{0}), y_{0}) → W_{(ξ_{0},η_{0}),y_{0}} = ℝ^{2} \ {(ξ_{0}, η_{0}) − x_{0}}

in (3.2) is a universal covering. This is visibly consistent with (3.3). In order to parametrize the space of leaves _{G}M, we observe that the parameter x_{0} can be eliminated. In fact, from the previous formulas we see that

(6.5)  L((ξ′_{0}, η′_{0}), (x′_{0}, u′)) = L((ξ_{0}, η_{0}), (x_{0}, u))

if and only if (ξ′_{0}, η′_{0}) − x′_{0} = (ξ_{0}, η_{0}) − x_{0} and u′ = ue^{α(θ_{0}−θ′_{0})}, so that we have z′(θ) = u′e^{αθ′_{0}} e^{−αθ(t)} = ue^{αθ_{0}} e^{−αθ(t)} = z(θ). In particular, it follows that

(6.6)  L((ξ_{0}, η_{0}), y_{0}) = L((ξ′_{0} + 1, η′_{0}), (1, 0, u′)),

where (ξ′_{0}, η′_{0}) = (ξ_{0}, η_{0}) − x_{0} , u′ = ue^{αθ_{0}} , θ′_{0} = 0, projecting to ℝ^{2} \ {(ξ′_{0}, η′_{0})}. Therefore the leaves of the form L((ξ_{0} + 1, η_{0}), (1, 0, u)) are distinct for different values of (ξ_{0}, η_{0}) and fixed value of u and from the relation (3.3) we conclude that

(6.7)  L((ξ_{0} + 1, η_{0}), (1, 0, u)) = (ξ_{0}, η_{0}) + L((1, 0), (1, 0, u)),

that is, G = ℝ^{2} acts without isotropy on _{G}M. We also need to determine the range for the parameter u. Obviously, we have L((1, 0), (1, 0, u′)) = L((1, 0), (1, 0, u)) if and only if u′ = e^{2παn}u for n ∈ ℤ. Thus these leaves are parametrized by [u], taking values in the quotient of the additive group ℝ under the multiplicative group {e^{2παn} : n ∈ ℤ}, that is,

(6.8)  {0} ∪ 𝕊_{+}^{1} ∪ 𝕊_{−}^{1} ≅ {0} ∪ ℝ_{+}^{×}/{e^{2παn} : n ∈ ℤ} ∪ ℝ_{−}^{×}/{e^{2παn} : n ∈ ℤ}.

The topology on the above space is determined by the leaf closures, respectively the orbit closures. First we have \overline{L((ξ_{0} + 1, η_{0}), (1, 0, u))} = (ξ_{0}, η_{0}) + \overline{L((1, 0), (1, 0, u))} in G × M, and it is sufficient to determine the closures of L((1, 0), (1, 0, u)). For (1, 0, u) ∈ M with u ≠ 0 we consider the curve c(θ) = e^{iθ} ∈ G = ℝ^{2}. It is liftable to G × M and determines on M the curve y(t) = (cos θ, sin θ, ue^{−αθ}). Thus the curve (c(θ), y(θ)) in the leaf through (1, 0; 1, 0, u) ∈ G × M ⊂ ℝ^{5} has a limit cycle for θ → ∞ which lies in the different leaf through (1, 0; 1, 0, 0), which is closed, given by the (x, y)-plane (ℝ^{2} × 0) \ 0 at level (1, 0) ∈ G. Thus we have

(6.9)  \overline{L((1, 0), (1, 0, u))} = L((1, 0), (1, 0, u)) ∪ L((1, 0), (1, 0, 0)).

Hence the leaf L((1, 0), (1, 0, u)) is not closed, and the topological space _{G}M is not T_{1} and not a manifold. The orbits of the 𝔤-action are determined by the leaf structure via pr_{2} in diagram (3.2), and they look here as follows. The (x, y)-plane (ℝ^{2} ×0)\0 is a closed orbit. Orbits above this plane are helicoidal staircases leading down and accumulating exponentially at the (x, y)-plane. Orbits below this plane are helicoidal staircases leading up and again accumulating exponentially. Thus the orbit space M/𝔤 of the 𝔤-action is given by (6.8), with the point 0 being closed. By (6.9), the closure of any orbit represented by a point [u] on one of the circles is given by {[u], 0}. From (6.6) and (6.7), we see that the G-completion _{G}M has a section over the orbit space _{G}M/G ≅ M/𝔤 given by [u] ↦ L((1, 0), (1, 0, u)). Therefore GM ≅ G × M/𝔤 = ℝ^{2} × { {0} ∪ 𝕊_{+}^{1} ∪ 𝕊_{−}^{1} }.

The structure of the completion and the orbit spaces are independent of the deformation parameter α > 0 in (6.1). However, for α ↓ 0, the completion just means adding in the z-axis, that is, we get _{G}M ≅ ℝ^{3} with G = ℝ^{2} acting by parallel translation on the affine planes z = c, and M/𝔤 ≅ _{G}M/G ≅ ℝ as it should be.

It was pointed out to us [2] that one can make this example still more pathological. Consider the above example only in a cylinder over the annulus 0 < x^{2}+y^{2} < 1. Add an open handle to the disk and continue the ℝ^{2}-action on the cylinder over the disk with an open handle added in such a way that there is a shift in the z-direction when one traverses the handle. Then one of the helicoidal staircases is connected to the the disk itself, so it accumulates onto itself. This is called a ‘resilient leaf’ in foliation theory.

7. Theorem. Let M be a connected transitive effective 𝔤-manifold. Let G be a connected Lie group with Lie algebra 𝔤. Then we have:

(7.1) There exists a subgroup H ⊂ G such that the G-completion _{G}M is diffeo-

morphic to G/H. (7.2) The Hausdorff quotient of _{G}M is the homogeneous manifold G/\overline{H}. It has

the following universal property: For each smooth 𝔤-equivariant mapping f : M → N into a Hausdorff G-manifold N there exists a unique smooth G-equivariant mapping f̃: G/\overline{H} → N with f = f̃∘ π ∘ j_{e} : M → G/H ⟶^{π} G/\overline{H} → N.

(7.3) For each leaf L(g, x_{0}) ⊂ G × M the projection pr_{2} : L(g, x_{0}) → M is a smooth fiber bundle with typical fiber H.

Proof. (7.1) We choose a base point x_{0} ∈ M. The G-completion is given by _{G}M = G ×_{𝔤} M, the orbit space of the 𝔤-action on G × M which is given by 𝔤 ∋ X ↦ L_{X} × ζ_{X}^{M}, and the G-action on the completion is given by multiplication from the left. The submanifold G × {x_{0}} meets each 𝔤-orbit in G × M transversely, since

T_{(g,x_{0})}(G × {x_{0}}) + T_{(g,x_{0})}L(g, x_{0}) = {L_{X}(g) × 0_{x_{0}} + L_{Y} (g) × ζ_{Y} (x_{0}) : X, Y ∈ 𝔤}
= T_{(g,x_{0})}(G × M).

By (3.3) we have L(g, x) = g.L(e, x) so that the isotropy Lie algebra 𝔥 = 𝔤_{x_{0}} = {X ∈ 𝔤 : ζ_{X}(x_{0}) = 0} is also given by

X ∈ 𝔥 ⟺ X × 0_{x_{0}} ∈ T_{(e,x_{0})}(G × {x_{0}}) ∩ T_{(e,x_{0})}L(e, x_{0})
⟺ L_{X}(g) × 0_{x_{0}} ∈ T_{(g,x_{0})}(G × {x_{0}}) ∩ T_{(g,x_{0})}L(g, x_{0}).

Since G × {x_{0}} is a leaf of a foliation and the L(e, x) also form a foliation, 𝔥 is a Lie subalgebra of 𝔤. Let H_{0} be the connected Lie subgroup of G which corresponds to 𝔥. Then clearly H_{0} × {x_{0}} ⊂ G × {x_{0}} ∩ L(e, x_{0}). Let the subgroup H ⊂ G be given by

H = {g ∈ G : (g, x_{0}) ∈ L(e, x_{0})} = {g ∈ G : L(g, x_{0}) = L(e, x_{0})};

then the C^{∞}-curve component of H containing e is just H_{0}. So H consists of at most countably many H_{0}-cosets. Thus H is a Lie subgroup of G (with a finer topology, perhaps). By construction the orbit space G ×_{𝔤} M equals the quotient of the transversal G × {x_{0}} by the relation induced by intersecting with each leaf L(g, x_{0}) separately, i.e., G ×_{𝔤} M = G/H.

(7.2) Obviously the T_{1}-quotient of G/H equals the Hausdorff quotient G/\overline{H}, which is a smooth manifold. The universal property is easily seen.

(7.3) Let x ∈ M and (g, x) ∈ L(e, x_{0}) = L(g, x) = g.L(e, x). So it suffices to treat the leaf L(e, x). We choose X_{1}, …, X_{n} ∈ 𝔤 such that ζ_{X_{1}}(x), …, ζ_{X_{n}}(x) form a basis of the tangent space T_{x}M. Let u : U → ℝ^{n} be a chart on M centered at x such that u(U) is an open ball in ℝ^{n} and such that ζ_{X_{1}}(y), …, ζ_{X_{n}}(y) are still linearly independent for all y ∈ U. For y ∈ U consider the smooth curve c_{y} : [0, 1] → U given by c_{y}(t) = u^{−1}(t.u(y)). We consider

∂_{t}c_{y}(t) = c′_{y}(t) = ∑_{i=1}^{n}f_{y}^{i}(t) ζ_{X_{i}}(c_{y}(t)), f_{y}^{i} ∈ C^{∞}([0, 1], ℝ),
X_{y}(t) = ∑_{i=1}^{n}f_{y}^{i}(t) X_{i} ∈ 𝔤,  X ∈ C^{∞}([0, 1], 𝔤),
g_{y} ∈ C^{∞}([0, 1], G), T (μ_{g_{y}(t)})∂_{t}g_{y}(t) = X_{y}(t), g_{y}(0) = e,

and everything is also smooth in y ∈ U. Then for h ∈ H we have (h.g_{y}(t), c_{y}(t)) ∈ L(e, x) since

∂_{t}(h.g_{y}(t), c_{y}(t)) = (L_{X_{y}(t)}(h.g_{y}(t)), ζ_{X_{y}(t)}(c_{y}(t))).

Thus U × H ∋ (y, h) ↦ pr_{2}^{−1}(U) ∩ L(e, x) is the required fiber bundle parameterization. □

8. Example. Let G be a simply connected Lie group and let H be a connected Lie subgroup of G which is not closed. For example, let G = Spin(5), which is compact of rank 2, and let H be a dense 1-parameter subgroup in its 2-dimensional maximal torus. Let Lie(G) = 𝔤 and Lie(H) = 𝔥. We consider the foliation of G into right H-cosets gH which is generated by {L_{X} : X ∈ 𝔥} and is left invariant under G. Let U be a chart centered at e on G which is adapted to this foliation, i.e. u : U → u(U) = V_{1} × V_{2} ⊂ ℝ^{k} × ℝ^{n−k} such that the sets u^{−1}(V_{1} × {x}) are the leaves intersected with U. We assume that V_{1} and V_{2} are open balls, and that U is so small that exp : W → U is a diffeomorphism for a suitable convex open set W ⊂ 𝔤. Of course, 𝔤 acts on U and respects the foliation, so this 𝔤-action descends to the leaf space M of the foliation on U which is diffeomorphic to V_{2}.

Lemma. In this situation, for the G-completion we have G ×_{𝔤} M = G/H.

Proof. We use the method described in the end of the proof of Theorem 7: _{G}M = G ×_{𝔤} M is the quotient of the transversal G × {x_{0}} by the relation induced by intersecting with each leaf L(g, x_{0}) separately. Thus we have to determine the subgroup H_{1} = {g ∈ G : (g, x_{0}) ∈ L(e, g)}.

Obviously any smooth curve c_{1} : [0, 1] → H starting at e is liftable to L(e, x_{0}) since it does not move x_{0} ∈ M. So H ⊆ H_{1}, and moreover H is the C^{∞}-path component of the identity in H_{1}.

Conversely, if c = (c_{1}, c_{2}) : [0, 1] → L(e, x_{0}) ⊂ G × M is a smooth curve from (e, x_{0}) to (g, x_{0}), then c_{2} is a smooth loop through x_{0} in M and there exists a smooth homotopy h in M which contracts c_{2} to x_{0}, fixing the ends. Since pr_{2} : L(e, x_{0}) → M is a fiber bundle by (7.3), we can lift the homotopy h from M to L(e, x_{0}) with starting curve c, fixing the ends, and deforming c to a curve c′ in L(e, x_{0}) ∩ pr_{2}^{−1}(x_{0}). Then pr_{1} ∘ c′ is a smooth curve in H_{1} connecting e and g.

Thus H_{1} = H, and consequently _{G}M = G/H. □

9. Theorem. Let M be a connected 𝔤-manifold. Let G be a connected Lie group with Lie algebra 𝔤. Then the G-completion _{G}M can be described in the following way:

(9.1) Form the leaf space M/𝔤, a quotient of M which may be non-Hausdorff and

not T_{1} etc. (9.2) For each point z ∈ M/𝔤, replace the orbit π^{−1}(z) ⊂ M by the homogeneous

space G/H_{x} described in Theorem 7, where x is some point in the orbit π^{−1}(z) ⊂ M. One can use transversals to the 𝔤-orbits in M to describe this in more detail.

(9.3) For each point z ∈ M/𝔤, one can also replace the orbit π^{−1}(z) ⊂ M by the homogeneous space G/\overline{H_{x}} described in Theorem 7, where x is some point in the orbit π^{−1}(z) ⊂ M. The resulting G-space has then Hausdorff orbits which are smooth manifolds, but the same orbit space as M/𝔤.

See Example 6 above.

Proof. Let O(x) ⊂ M be the 𝔤-orbit through x, i.e., the leaf through x of the singular foliation (with non-constant leaf dimension) on M which is induced by the 𝔤-action. Then the G-completion of the orbit O(x) is _{G}O(x) = G/H_{x} for the Lie subgroup H_{x} ⊂ G described in Theorem (7.1). By the universal property of the G-completion we get a G-equivariant mapping _{G}O(x) → _{G}M which is injective and a homeomorphism onto its image, since we can repeat the construction of Theorem (7.1) on M. Clearly the mapping j_{e} : M → _{G}M induces a homeomorphism between the orbit spaces M/𝔤 → _{G}M/G.

Now let s : V → M be an embedding of a submanifold which is a transversal to the 𝔤-foliation at s(v_{0}). We have T s · T_{v_{0}}V ⊕ ζ_{s(v_{0})}(𝔤) = T_{s(v_{0})}M. Then s induces a mapping V → G × M and V → _{G}M and we may use the point s(v) in replacing O(s(v)) by G/H_{s(v)} for v near v_{0}. □

The following diagram summarizes the relation between the preceding constructions.

(9.4)  [figure: M; =; ⋃_{[x]∈M/𝔤} G/H_{x}; ≅; ⋃_{[x]∈M/𝔤} G/\overline{H_{x}}; M; π; j_{e}; _{G}M = G ×_{𝔤} M; π_{G}; G × M/\overline{F}_{ζ}; π̄_{G}; M/𝔤; ≅; _{G}M/G; (G × M/\overline{F}_{ζ})/G]

Note that taking the T_{1}-quotient G × M/\overline{F}_{ζ} of the leaf space _{G}M may be a very severe reduction. In Example 6 the isotropy groups H_{x} are trivial and we have G × M/\overline{F}_{ζ} = ℝ^{2} × \{0\} and (G × M/\overline{F}_{ζ})/G = \{0\}.

10. Palais’ treatment of 𝔤-manifolds. In [7], Palais considered 𝔤-actions on finite-dimensional manifolds M in the following way. He assumed from the beginning that M may be a non-Hausdorff manifold, since the completion may be non-Hausdorff. Then he introduced notions which we can express as follows in the terms introduced here:

(10.1) (M, ζ) is called generating if it generates a local G-transformation group. See [7, II, 2, Def. V and II, 7, Thm. XI]. This holds if and only if the leaves of the graph foliation on G × M described in Section 3 are Hausdorff. For Hausdorff 𝔤-manifolds this is always the case.

(10.2) (M, ζ) is called uniform if pr_{1} : L(e, x) → G in (3.2) is a covering map for each x ∈ M. See [7, III, 6, Def. VIII and III, 6, Thm. XVII, Cor., Cor. 2]. In the Hausdorff case the 𝔤-action is then complete and it may be integrated to a G̃-action, where G̃ is a simply connected Lie group with Lie algebra 𝔤, so that G̃M ≅ M.

(10.3) (M, ζ) is called univalent if pr_{1} : L(e, x) → G in (3.2) is injective for all x.

See [7, III, 2, Def. VI and III, 4, Thm. X]. (10.4) (M, ζ) is called globalizable if there exists a (non-Hausdorff) G-manifold N

which contains M equivariantly as an open submanifold. See [7, III, 1, Def. II and III, 4, Thm. X]. This is a severe condition which is not satisfied in examples 4 and 6 above.

Palais’ main result on (non-Hausdorff) manifolds with a vector field says that (10.1), (10.3), and (10.4) are equivalent. See [7, III, 7, Thm. XX].

On (non-Hausdorff) 𝔤-manifolds his main result is that (10.3) and (10.4) are equivalent. See [7, III, 1, Def. II and III, 4, Thm. X], and also [7, III, 2, Def. VI and III, 4, Thm. X].

11. Concluding remarks. (11.1) A suitable setting for further development might be the class of discrete 𝔤-manifolds, that is, 𝔤-manifolds for which the G̃- space G̃M is T_{1}, or equivalently the leaves of the graph foliation F_{ζ} on G̃ × M are closed. In this case, the charts j_{g} : M → G̃M in (5.1) are local diffeomorphisms with respect to the unique smooth structure on G̃M, and G̃M is a smooth manifold, albeit not necessarily Hausdorff.

(11.2) In the context of (11.1), there are several definitions of proper 𝔤-actions, all of which are equivalent to saying that the G̃-action on G̃M is proper. Many properties of proper actions will carry over to this case.

References

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4. Boris Khesin and Peter W. Michor, The flow completion of Burgers’ equation, pp. 1–8, Walter de Gruyter, Berlin, 2004, IRMA Lectures in Mathematics and Theoretical Physics. http://www.mat.univie.ac.at/˜ michor/burgers.ps

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Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801

   E-mail address: kamber@math.uiuc.edu

Institut für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria, and Erwin Schrödinger Institut für Mathematische Physik, Boltzmanngasse 9, A-1090 Wien, Austria

   E-mail address: michor@esi.ac.at