Symmetry, Integrability and Geometry: Methods and Applications The Infinitesimalization and Reconstruction of Locally Homogeneous Manifolds

A linear connection on a Lie algebroid is called a Cartan connection if it is suitably compatible with the Lie algebroid structure. Here we show that a smooth connected manifold $M$ is locally homogeneous - i.e., admits an atlas of charts modeled on some homogeneous space $G/H$ - if and only if there exists a transitive Lie algebroid over $M$ admitting a flat Cartan connection that is 'geometrically closed'. It is shown how the torsion and monodromy of the connection determine the particular form of $G/H$. Under an additional completeness hypothesis, local homogeneity becomes global homogeneity, up to cover.


Introduction
Let M be a smooth connected manifold. Then M is locally homogeneous if, for some homogeneous space G/H, the smooth structure of M can be stiffened to a G-structure, where G is the pseudogroup of all those local transformations of G/H that are restrictions of a left translation by an element of G.

Main results
The chief purpose of this article is to re-examine local homogeneity from the Lie groupoid point of view. This leads, in particular, to the conclusion that a locally homogeneous manifold can be infinitesimalized to obtain a transitive Lie algebroid over M , equipped with a flat linear connection ∇ that is compatible with the Lie algebroid structure, i.e. is a flat Cartan connection (see Section 2.1 below). As we shall elucidate, ∇ being flat amounts to the existence of a transitive action by some Lie algebra on M 'twisted' by a monodromy representation.
Not all transitive Lie algebroids equipped with a flat Cartan connection are infinitesimalizations of a locally homogeneous manifold. If (g, ∇) is an infinitesimalization, then ∇ must be geometrically closed, in a sense made precise below. Fortunately, geometric closure is sufficient for reversing the infinitesimalization procedure: As we show in Section 5, the particular model G/H that applies is encoded in the torsion and monodromy of ∇. By contrast, in the predominant approach to local homogeneity, one fixes a particular homogeneous space G/H a priori, and asks if M admits G/H as a local model. In practice, this requires one to anticipate an appropriate model, or check several candidates systematically. See, e.g., the survey [10] for this point of view. arXiv:1304.7838v3 [math.DG] 26 Nov 2013

Geometric closure
Let g be a transitive Lie algebroid equipped with a flat Cartan connection ∇. Fix an arbitrary point m 0 ∈ M and an arbitrary simply-connected open neighbourhood U of m 0 . Let g 0 be the finite-dimensional vector space of ∇-parallel sections over U . Because ∇ is flat, g 0 is the same for all choices of U , up to obvious identifications. Because ∇ is Cartan, g 0 ⊂ Γ(g U ) is a subalgebra (with bracket encoded in the torsion of ∇; see Section 2.5).
Denote the simply-connected Lie group having g 0 as its Lie algebra by G 0 , and let h 0 denote the kernel of the map ξ → #ξ(m 0 ) : g 0 → T m 0 M.
Here # : g → T M denotes the anchor of g. It is easy to see that h 0 ⊂ g 0 is a subalgebra, and we say ∇ is geometrically closed if the connected subgroup H 0 ⊂ G 0 with Lie algebra h 0 is closed in G 0 . In making this definition, the choice of fixed point m 0 ∈ M is immaterial, as we will establish later in Section 4.1.
The basic prototype of a Lie algebroid supporting a flat Cartan connection is the action algebroid g 0 × M associated with some infinitesimal action of a Lie algebra g 0 on M (see Section 2.3). Indeed, locally this is the only example (see Section 2.5). In this case, h 0 ⊂ g 0 is the isotropy subalgebra at m 0 .
For an example a Lie algebra action that is not geometrically closed, see [12, Example 8] and [13].

Discussion
The novelty of Theorem 1.1 lies mainly in the point of view, as explained in Section 1.1 above. The implications of this change in viewpoint, as it applies to other parts of Cartan's generalization of the Klein Erlangen program, are explored in [4]. In its contemporary conception, this program is described in, e.g., [16].
A Lie algebroid over M equipped with a flat Cartan connection amounts to a Lie algebra action on the universal coverM that suitably respects covering transformations. We reconstruct a locally homogeneous structure on M by applying Cartan's development technique to such actions. Excellent expositions of this technique may be found elsewhere; see, e.g., [16]. That said, Dazord's integrability result [8], and Lie groupoid formalism, provide for an economic treatment of development, which is offered in Section 4. For a detailed treatment of infinitesimal actions of Lie algebras, we refer the reader to [2,12].
It should be noted that the Lie group G occurring in Theorem 1.1 (and in Theorem 1.2 mentioned below) need not be connected. Actually, one may insist that G be connected (indeed simply-connected) but only at the cost of allowing transition functions more general than left translations: rather they may be arbitrary affine transformations (for a definition, see Section 2.4). Our proof of Theorem 1.1 begins with the proof of a variant along these lines. Theorem 1.1 suggests a two-step strategy for establishing the local homogeneity of a smooth manifold: (i) construct a transitive Lie algebroid g over M , equipped with some Cartan connection ∇ (not necessarily flat); and (ii) attempt to deform ∇, within the class of Cartan connections, to one that is simultaneously flat and geometrically closed.
For example, associated with any Riemann surface M is a canonical Lie algebroid g ⊂ J 1 (T M ) (the 'isotropy' of the complex structure; see [4]). Associated with any compatible metric σ is a subalgebroid g σ ⊂ g. Applying Cartan's method of equivalence, as we have described in [4], one constructs a Cartan connection ∇ on g σ , which extends rather naturally to one on g. Applying step (ii) above to ∇, one can establish the existence of an atlas of affine coordinates when M has vanishing Euler characteristic χ; applying (ii) to a 'prolongation' ∇ (1) of ∇ -a connection on the prolongation g (1) ⊂ J 2 (T M ) of g -one obtains an atlas of complex projective coordinates, provided χ > 1. That is, we recover a weak version of the uniformization theorem, as in [11]. Details will appear elsewhere.
In any case, Theorem 1.1 furnishes a strategy for constructing topological invariants of smooth manifolds, arising as obstructions to the existence of locally homogeneous structures. Theorem 1.1 also engenders the following question: What are the global analogues of intransitive Lie algebroids over M equipped with flat Cartan connections? The answer: A class of differentiable pseudogroups of transformations on M , generalizing the canonical pseudogroups of transformations associated with locally homogeneous structures. As it turns out, such pseudogroups are not necessarily Lie pseudogroups in the classical sense. They nevertheless have a very satisfactory 'Lie theory' which is sketched in [4, Appendix A] and described further in [5].

Completeness and homogeneity up to cover
Suppose there exists a Lie group G acting transitively on the universal coverM of M , with G understood to contain the group Γ ∼ = π 1 (M ) of covering transformations as a subgroup, and with the action of G onM extending the tautological action of Γ. In this case M is a double quotient of groups, M ∼ = (G/H)/Γ, and M may be said to be homogeneous up to cover. Under a suitable completeness hypothesis, locally homogeneous manifolds are already homogeneous up to cover. See, for example, Thurston's lucid account [17].
To formulate a notion of completeness for Cartan connections leading to a strengthening of Theorem 1.1, let g be any Lie algebroid over M and let t → X t ∈ g be a smooth path, defined on some interval of the real line. Let m t ∈ M be the footprint of X t andṁ t ∈ T M its velocity. Then t → X t is called a g-path if #(X t ) =ṁ t , for all t. If ∇ is a linear connection on g (not necessarily flat or Cartan), then a g-path X t ∈ g is called a geodesic of ∇ if ∇ṁ t X t = 0 for all t. A geodesic of the Levi-Cevita connection associated with a Riemannian metric is then a geodesic in the standard sense if we take g = T M .
The usual argument for the existence of geodesics in Riemannian geometry carries over to the general case: Through every point of g there passes a unique geodesic of ∇. We call ∇ complete if every geodesic X t of ∇ can be defined for all time t ∈ R.
The reader is to be warned that compactness of M is not sufficient for completeness of ∇; a simple counterexample is given in Section 6.1. If, in addition, the image of the monodromy representation has compact closure, then ∇ is indeed complete. Alternatively, if M admits a complete Riemannian metric invariant with respect to a natural representation of the Lie algebroid g on S 2 (T * M ), then ∇ is again complete. Precise statements and proofs are given in Section 6, along with a proof of the following variation of Theorem 1.

Paper outline
The present article is organized as follows: In Section 2 we review the notion of Cartan connections on Lie algebroids and explain the sense in which flat Cartan connections amount to Lie algebra actions twisted by a monodromy representation (Theorem 2.7). In Section 3, we construct the infinitesimalization (g, ∇) of a locally homogeneous structure on M and observe that it is geometrically closed. This establishes the necessity of the conditions in Theorem 1.1.
In Section 4 we define the development of the infinitesimal action of a Lie algebra and describe its behavior under 'equivariant coordinate changes'. This is applied in Section 5 to reconstruct a locally homogeneous structure from any transitive Lie algebroid equipped with a flat, geometrically closed, Cartan connection, which completes the proof of Theorem 1.1.
In Section 5 we also explain how the torsion and monodromy of ∇ determine the particular homogeneous model G/H that applies.
In Section 6, we prove Theorem 1.2 and offer some sufficient conditions for completeness. The last section, Section 7, illustrates our results by characterising complete local Lie groups, and by recovering a variant of the well-known classification theorem for complete Riemann manifolds.

Notation
Throughout this paper g denotes a Lie algebroid, g 0 and h 0 Lie algebras, G 0 , H 0 , G and H Lie groups, and G a Lie groupoid.

Cartan connections on Lie algebroids
We assume the reader is familiar with the notion of g-connections and g-representations (g a Lie algebroid). See, for example, [3] or [4]. With the exception of Section 2.6, the present section is mostly a summary of [3].

Cartan connections def ined
Let g be a vector bundle over M . Then there is a one-to-one correspondence between linear connections ∇ on g and splittings s ∇ : g → J 1 g of the associated exact sequence this correspondence is given by Here J 1 g is the vector bundle of one-jets of sections of g, and the inclusion T * M ⊗ g → J 1 g is the morphism whose corresponding map on section spaces sends df ⊗ X to f J 1 X − J 1 (f X). Now suppose that g is not just a vector bundle but a Lie algebroid. Then ∇ is a Cartan connection if the vector bundle morphism s ∇ : g → J 1 g is a morphism of Lie algebroids.
Recall here that J 1 g has a natural Lie algebroid structure determined by the requirement Here and throughout, the anchor of a Lie algebroid is denoted #. For details and an explicit formula for the bracket on J 1 g, see [3], where it is also shown that ∇ is Cartan if and only if its cocurvature vanishes. The latter is a tensor cocurv ∇ ∈ Γ( In the above formula∇ denotes the so-called associated g-connection on T M , defined bȳ ∇ X V = #∇ V X + [#X, V ]. There is also an associated g-connection on g itself, also denoted∇, and defined by∇ X Y = ∇ Y X + [X, Y ]. For this connection one can define torsion in the usual way, by When ∇ is Cartan both associated connections are flat, i.e., define representations of the Lie algebroid g on T M and g.

Cartan connection-preserving morphisms
From a well-known characterization of Lie algebroid morphisms given in, e.g., [14,Proposition 4.3.12], one readily establishes the following: Proposition 2.1. Let g 1 be a Lie algebroid over M 1 with Cartan connection ∇ 1 and g 2 a Lie algebroid over M 2 with Cartan connection ∇ 2 . Then a connection-preserving vector bundle map Φ : g 1 → g 2 , covering some smooth map φ : M 1 → M 2 , is a Lie algebroid morphism if and only if:
Here∇ i is the associated g i -connection on g i (see Section 2.1).

Equivariance with twist
Let g 0 be a finite-dimensional Lie algebra. If g 0 acts smoothly on M from the left 1 then we denote the corresponding Lie algebra homomorphism g 0 → Γ(T M ) by ξ → ξ † . The canonical flat connection ∇ on the action algebroid g = g 0 × M is an example of a Cartan connection. As we recall in Section 2.5 below, this is, locally, the only example of a flat Cartan connection.
Recall that the anchor of an action algebroid g 0 × M is defined by #(ξ, m) = ξ † (m), and that the bracket on sections of g 0 × M (g 0 -valued functions on M ) is given by Note that the associated g-connection∇ on g has torsion τ . Now suppose that g 0 acts smoothly on two manifolds M 1 and M 2 , and let End(g 0 ) denote the vector space of Lie algebra endomorphisms of g 0 . Then we will say that a smooth map φ : M 1 → M 2 is g 0 -equivariant with twist µ ∈ End(g 0 ) if ξ † and (µξ) † are φ-related, for all ξ ∈ g 0 . The twist need not be unique. Applying Proposition 2.1, we obtain: Proposition 2.2. Every connection-preserving vector bundle morphism The significance of g 0 -equivariance in the present context is established in Theorem 2.7 below.

Af f ine transformations
Let g 0 act on M as above, and let G 0 be the simply-connected Lie group integrating g 0 . As we show in Section 4.3, if the action of g 0 is transitive and geometrically closed, then a g 0 -equivariant We pause now to define and characterise such maps.
For every Lie group G 0 one has the group Aff(G 0 ) ⊂ Diff(M ) of affine transformations, generated by left translations, right translations, and group automorphisms. (Affine transformations of the Abelian group R n are then affine transformations of R n in the standard sense of the term.) If H 0 ⊂ G 0 is a subgroup, then some elements of Aff(G 0 ) descend to bijections of G 0 /H 0 that we also refer to as affine transformations, forming a group denoted by Aff(G 0 /H 0 ). All left translations in G 0 descend to elements of Aff(G 0 /H 0 ); a right translation by k ∈ G 0 descends if and only if k is in the normaliser of H 0 ; more generally, an element of Aff(G 0 /H 0 ) is of the form g → kΨ(g) for some k ∈ G 0 and Ψ ∈ Aut(G 0 ), and descends to an element of Aut On the other hand, we say that a smooth map φ : The magnanimous reader will readily verify the following characterization: , if and only if it is affine.

The local form of a Lie algebroid with f lat Cartan connection
For the moment, suppose that g is an arbitrary vector bundle over M , equipped with a linear connection ∇. Denote the finite-dimensional subspace of ∇-parallel sections of g by g 0 . In the special case that ∇ is flat, and M is simply-connected, we obtain a connection-preserving isomorphism Proposition 2.4. Let g be a Lie algebroid over a smooth connected manifold M and ∇ a Cartan connection, not necessarily flat. Then: (1) The subspace g 0 ⊂ Γ(g) of ∇-parallel sections is a Lie subalgebra.
(2) The bracket on g 0 coincides with the torsion of the associated g-connection on g, in the sense that for any m ∈ M ; ξ, η ∈ g 0 . Here∇ denotes the associated g-connection on g.
(3) The mapping (ξ, m) → #ξ(m) : g 0 ×M → T M defines a smooth action of the Lie algebra g 0 on M , making g 0 × M into an action algebroid.
(4) If ∇ is flat, and M is simply-connected, then the canonical isomorphism g ∼ = g 0 × M in (2.2) is an isomorphism of Lie algebroids.

Monodromy and the global form
Again suppose that g is an arbitrary vector bundle over M . Letg denote its pullback to a vector bundle over the universal coverM of M . Let Γ denote the group of covering transformations of M . Then it is an elementary observation that there exists a unique lift of the tautological action of Γ onM to an action ong satisfying the following requirement: For all X ∈g and φ ∈ Γ, X and φ · X have the same image under the canonical projectiong → g. Evidently Γ acts ong by vector bundle automorphisms, and we recover the original vector bundle over M as a quotient: g =g/Γ. Now suppose g is equipped with a flat linear connection ∇ and let∇ denote the pullback of ∇ to a connection ong, also flat. Let g 0 ⊂ Γ(g) denote the vector space of∇-parallel sections. Then, asM is simply-connected, we have a canonical isomorphismg ∼ = g 0 ×M (as discussed in Section 2.5 above) and consequently g ∼ = (g 0 ×M )/Γ. Since the action of Γ ong described above automatically preserves∇, it must be of the form for some uniquely determined group homomorphism µ → µ φ : Γ → Aut(g 0 ). This is the monodromy representation associated with the flat connection ∇.
Remark 2.5. As the reader will recall, monodromy has the following alternative interpretation. Letm 0 ∈M be a point covering any fixed point m 0 ∈ M . Then there is an isomorphism g 0 ∼ = g| m 0 in which each ξ ∈ g 0 corresponds to the image of ξ(m 0 ) under the canonical projectiong → g. Also, Γ may be identified with the fundamental group π 1 (M, m 0 ). Under these identifications µ φ (ξ) is the ∇-parallel translate of ξ ∈ g| m 0 along any closed path γ in M representing φ ∈ π 1 (M, m 0 ). Or, this parallel translation may be viewed in the following way: each ξ ∈ g| m 0 extends to a locally defined ∇-parallel section X of g, which can be 'analytically continued' around γ and re-evaluated at m 0 to obtain µ φ (ξ).
Proposition 2.6. In the scenario above, suppose that g is a Lie algebroid and ∇ a flat Cartan connection on g, so that g 0 is a Lie algebra acting onM (see Proposition 2.4). Then: (1) The canonical isomorphism g ∼ = (g 0 ×M )/Γ is an isomorphism of Lie algebroids.
(2) The group of covering transformations Γ acts in the monodromy representation µ → µ φ by Lie algebra automorphisms of g 0 .
(4) If we restrict the tensor tor∇ ∈ Γ(∧ 2 (g * ) ⊗ g) to define a bracket on the fibre g| m 0 , then the isomorphism g 0 ∼ = g| m 0 in Remark 2.5 is an isomorphism of Lie algebras. Here∇ denotes the g-connection on g associated with ∇.
Proof . Evidently, there is a unique Lie algebroid structure ong such that the projectiong → g is a Lie algebroid morphism. With respect to this structure, the action of Γ ong is by Lie algebroid isomorphisms. Thus the isomorphism g ∼ =g/Γ may be regarded as a Lie algebroid isomorphism. (None of these statements depend on the existence of a trivializationg ∼ = g 0 ×M .) On the other hand, the isomorphismg ∼ = g 0 ×M determined by the flat connection∇ is a Lie algebroid isomorphism, by Proposition 2.4(4). So (1) holds.
The constructions above are reversible. Indeed, let g 0 be an arbitrary Lie algebra acting smoothly on the universal coverM of M , and let φ → µ φ : Γ → Aut(g 0 ) be a representation of Γ by Lie algebra automorphisms of g 0 satisfying (3) (of which there may be more than one). Then the action of Γ on the action algebroidg := g 0 ×M defined by (2.3) is by Lie algebroid automorphisms (by Proposition 2.2), implying that the quotient g :=g/Γ is a Lie algebroid. Moreover, the canonical flat Cartan connection ong = g 0 ×M drops to a flat Cartan connection on g whose monodromy is precisely µ. This establishes the following: Theorem 2.7. Let Γ ∼ = π 1 (M ) denote the group of covering transformations ofM . Then there is a natural one-to-one correspondence between: (i) pairs (g, ∇), where g is a Lie algebroid over M and ∇ is a flat Cartan connection; and (ii) pairs (g 0 , µ), where g 0 is a finite-dimensional Lie algebra acting smoothly onM , and µ is a representation of Γ on g 0 by Lie algebra automorphisms such that each covering transformation φ ∈ Γ is a g 0 -equivariant diffeomorphism with twist µ φ ∈ Aut(g 0 ).

Inf initesimalization
In this section we prove necessity of the conditions in Theorem 1.1, i.e., for every locally homogeneous manifold M , there exists a transitive Lie algebroid g over M supporting a flat, geometrically closed, Cartan connection ∇. We refer to the particular pair (g, ∇) constructed below as the locally homogeneous structure's infinitesimalization.

Construction of the inf initesimalization
Suppose M is a locally homogeneous manifold modeled on G/H and let ψ i : U i → G/H, i ∈ I, be an atlas of coordinate charts adapted to the model. This means that for each i, j ∈ I for which U i ∩ U j = ∅, we are given an element g ji ∈ G, such that ψ j • ψ −1 . Moreover, the elements g ji satisfy the cocycle conditions g ii = id, and g ij g jk = g ik for all i, j, k ∈ I. Now let g 0 denote the Lie algebra of G. Infinitesimalizing the action of G on G/H, we obtain a Lie algebra homomorphism ξ → ξ † : g 0 → Γ(T (G/H)), which makes g 0 ×G/H into a transitive action algebroid. In particular, each restriction g 0 × ψ i (U i ) ⊂ g 0 × G/H is a transitive action algebroid over ψ i (U i ). We claim that the Lie algebroids g 0 ×ψ i (U i ), i ∈ I, are local trivializations of a single transitive Lie algebroid g over M , the canonical flat connections on the g 0 × ψ i (U i ) representing a flat Cartan connection ∇ on g.
To see this, define an equivalence relation ∼ on the set (the disjoint union of the sets g 0 ×ψ i (U i ), i ∈ I) by declaring (ξ, x, i) ∼ (ξ , x , i ) whenever there is an m ∈ U i ∩U i such that ψ i (m) = x, ψ i (m) = x and ξ = Ad g i i ξ. Then the set g of equivalence classes is a smooth vector bundle over M , with footprint projection [ξ, x, i] → ψ −1 i (x); here [ξ, x, i] denotes the class with representative (ξ, x, i).
The vector bundle g admits local trivializations given by with transition functions given by In particular, the transition functions preserve the canonical flat connections on the action algebroids g 0 × ψ i (U i ). It follows that there is a (necessarily flat) connection ∇ on g that is locally represented by the canonical flat connection on each g 0 × ψ i (U i ).
If we write L g (x) := g ·x, then the pushforward of ξ † by the transformation L g : G/H → G/H is (Ad g ξ) † . Using this fact, one easily sees that there is a well-defined vector bundle epimorphism # : g → T M , defined locally by It remains to define a Lie bracket on sections of g for which # is a compatible anchor. To this end, notice that each local trivialization Ψ i identifies a fibre g| m (m ∈ U i ) with g 0 . This identification depends on the local trivialization chosen, but only up to adjoint transformations of g 0 ; it consequently transfers the Lie bracket on g 0 to one on g| m that is trivialization-independent. We let τ ∈ ∧ 2 (g * ) ⊗ g denote the tensor whose restriction to each fibre g| m is the Lie bracket just defined. A bracket on g is then given by (cf. (2.1)): With this bracket g becomes a transitive Lie algebroid and the local trivializations Ψ i : g| U i → g 0 × ψ i (U i ) become connection-preserving Lie algebroid morphisms. In particular, the connection ∇ is Cartan and geometrically closed because its local representatives are.

The development of Lie algebra actions
To reconstruct a locally homogeneous structure, from a pair (g, ∇) satisfying the conditions in Theorem 1.1, we will use Cartan's development technique. We pause here to describe development using the economy afforded by Lie groupoid language, and to show that development is suitably equivariant. Before doing so, we argue that geometric closure, as defined in Section 1.2, is independent of the choice of fixed point m 0 ∈ M .

Geometric closure recharacterized
Evidently a flat Cartan connection ∇ on a transitive Lie algebroid g over M is geometrically closed 'at the point m 0 ∈ M ' if and only if∇ is geometrically closed at some pointm 0 ∈M covering m 0 . Here (g,∇) denotes the lift of (g, ∇) to the universal coverM , as described in Section 2.6. So, without loss of generality, we now suppose M is simply-connected. According to Proposition 2.4(4), g is isomorphic to an action algebroid g 0 × M , where g 0 has the same meaning as in Section 1.2. According to Dazord [8], all action algebroids are integrable. So there is a Lie groupoid G integrating g 0 ×M , which we may take to be source-simply-connected [7, Lie I].
Fix some m 0 ∈ M and let G m 0 m 0 denote the isotropy group at m 0 (the group of arrows of G simultaneously beginning and ending at m 0 ). Proof . Since we assume g 0 acts transitively, G is a transitive Lie groupoid (because its orbits are disjoint and open and M is connected). Consequently, if P ⊂ G denotes the source-fibre over m 0 (the subset of all arrows in G beginning at m 0 ), then P is a principal G m 0 m 0 -bundle over M , with the Lie group G m 0 m 0 acting on P from the right. The bundle projection P → M is just the restriction of the target-projection G → M of the groupoid G. For this principal bundle we have a corresponding long exact sequence in homotopy, Since P is connected and M is simply-connected, π 0 (G m 0 m 0 ) is trivial.
Let G 0 and H 0 have the meanings given in Section 1.2. Then then there exists a Lie groupoid morphism Ω : G → G 0 integrating the canonical projection g 0 ×M → g 0 (a morphism of Lie algebroids) [7, Lie II]. Evidently the subgroups Ω(G m 0 m 0 ) and H 0 of G 0 have the same Lie algebra h 0 . By the lemma they must coincide. This proves:

Development def ined
Let g 0 be any finite-dimensional Lie algebra acting smoothly on M . Assume the action is transitive and geometrically closed. The development is always defined as a map from the universal coverM (on which g 0 acts also) so, without loss of generality, we suppose once more that M is simply-connected. We again denote by G 0 the simply-connected Lie group having g 0 as Lie algebra, and let G, G m 0 m 0 , P , and Ω : G → G 0 have the meanings given in the preceding Section 4.1. By Proposition 4.2 and geometric closure, H 0 := Ω(G m 0 m 0 ) is a closed subgroup of G 0 , so that G 0 /H 0 is a smooth Hausdorff manifold.
Because Ω : G → G 0 integrates the projection g 0 × M → g 0 (a point-wise isomorphism), its restriction to P is a local diffeomorphism Ω : P → G 0 . Using the fact that Ω is a groupoid morphism, we see that Ω sends orbits of G m 0 m 0 in P to orbits of H 0 in G 0 (left cosets). Since P/G m 0 m 0 ∼ = M , it follows that Ω : P → G 0 descends to a map D : M → G 0 /H 0 :  Proof . This is a straightforward consequence of the fact that the principal bundle projection P → M is a surjective submersion (and so admits local sections), Ω : P → G 0 is a local diffeomorphism, and dim(G m 0 m 0 ) = dim(H 0 ).
Restricting the development to sufficiently small open sets in M , we obtain an atlas of charts trivially adapted the homogeneous space G 0 /H 0 : Corollary 4.4. If a finite-dimensional Lie algebra g 0 acts on a simply-connected manifold M , and this action is transitive and geometrically closed, then M is locally homogeneous. Indeed in that case there exists a closed subgroup H 0 ⊂ G 0 of the simply-connected Lie group integrating g 0 , and an atlas of charts adapted to the homogeneous model G 0 /H 0 whose transition functions are all identity transformations.

Behavior under equivariant coordinate changes
In order to describe how development transforms under equivariant coordinate changes, we wish to associate, with each g 0 -equivariant diffeomorphism φ : M → M with twist µ ∈ Aut(g 0 ), a corresponding diffeomorphism To this end, we require the following: Lemma 4.5. Let φ : M → M be any smooth g 0 -equivariant map with twist µ ∈ End(g 0 ). Let µ : G 0 → G 0 denote the unique group homomorphism with derivative µ, and let (µ × φ) ∧ : G → G denote the unique Lie groupoid morphism with derivative µ × φ : g 0 × M → g 0 × M . Then the following diagram commutes: Proof . The Lie groupoid morphisms Ω • (µ × φ) ∧ andμ • Ω have the same derivatives, namely (ξ, m) → µξ : g 0 × M → g 0 . By the uniqueness part of the generalization to Lie groupoids of Lie's second integrability theorem [7, Lie II], these morphisms must coincide.
Proof . Because, in the notation of Lemma 4.5, the groupoid automorphism (φ × µ) ∧ : G → G covers φ : M → M , and because q ∈ G is an arrow from m 0 to φ(m 0 ), we have Applying that lemma, we computê Lemma 4.6 implies that the map φ G 0 /H 0 : G 0 /H 0 → G 0 /H 0 , given implicitly by is well-defined. It is also independent of the choice of arrow q ∈ G from m 0 to φ(m 0 ), because Ω(q)H 0 = D(φ(m 0 )). Note that despite our choice of notation, φ G 0 /H 0 depends not just on φ but also on the twist µ.
(3) The following diagram commutes: Proof . Let q, q ∈ G be arrows from m 0 to φ(m 0 ), φ (m 0 ) respectively. Then q := (φ×µ) ∧ (q ) q is an arrow from m 0 to φ(φ (m 0 )). It follows that for an arbitrary element gH 0 ∈ G 0 /H 0 , we have where at the beginning of the second line we have applied Lemma 4.5. On the other hand, we have Comparing this equation with the preceding one establishes (1).
One deduces (2) immediately from the definition of φ G 0 /H 0 . Regarding (3), let m ∈ M be arbitrary and let p ∈ G be an arrow from m 0 to m. Then p := (φ × µ) ∧ (p)q is an arrow from m 0 to φ(m). Consequently, we compute At the beginning of the second line we have again applied Lemma 4.5.

Reconstruction
In this section we complete the proof of Theorem 1.1 by reconstructing a locally homogeneous structure from any transitive Lie algebroid supporting a flat, geometrically closed Cartan connection. We also explain how torsion and monodromy of the connection determine a suitable model G/H.

The role of torsion and monodromy
According to the following result, if M satisfies the hypotheses of Theorem 1.1, then it locally homogeneous in the apparently weaker sense of admitting an atlas in which the transition functions are affine transformations of a homogeneous model G 0 /H 0 (in the sense of Section 2.4). The advantage of this formulation over the one in Theorem 1.1 is that we may take G 0 to be simply-connected and H 0 to be connected.
Proposition 5.1. Let g be a transitive Lie algebroid, over a smooth connected manifold M , and ∇ a flat Cartan connection on g. Define a g-connection∇ on g bȳ and let tor∇ denotes its torsion: Then, fixing a point m 0 ∈ M , we have: (1) The restriction of tor∇ to g 0 := g| m 0 is a Lie bracket on g 0 .
Next, let Γ = π 1 (M, m 0 ) denote the fundamental group of M and let µ : Γ → g 0 denote the monodromy representation associated with the flat connection ∇. Then: (2) Γ acts on g 0 by Lie algebra automorphisms. In particular, the monodromy representation integrates to a group homomorphismμ : Γ → Aut(G 0 ), where G 0 denotes the simplyconnected Lie group with Lie algebra g 0 . Now let h 0 ⊂ g 0 = g| m 0 denote the kernel of the restriction of the anchor # : g → T M to g| m 0 , and let H 0 ⊂ G 0 denote the connected subgroup with Lie algebra h 0 . Then: (3) The connection ∇ is geometrically closed if and only if H 0 ⊂ G 0 is closed, in which case there exists a smooth action (4) There is an atlas of charts on M with model G 0 /H 0 such that each transition function is a restriction of some φ G 0 /H 0 , φ ∈ Γ.
Remark 5.2. In the proposition statement (but not the subsequent proof) monodromy is to be understood in the usual sense of parallel translation (or analytic continuation) around closed paths; see the Remark 2.5.
Before turning to the proof, let us explain how to recover genuine local homogeneity from the conclusion of the proposition and hence complete the proof of Theorem 1.1. Indeed, let G = Γ ×μ G 0 denote the semidirect product, with multiplication given by Then, as As this action is also transitive, we have a canonical isomorphism G 0 /H 0 ∼ = G/H, where H is the isotropy at idH 0 ∈ G 0 /H 0 of the action by G just defined. Viewing each transition function as a local transformation of G/H, it becomes a left-translation by some element of Γ ∼ = Γ × {id} ⊂ G.

Reconstruction
To prove the proposition, pull g back to a transitive Lie algebroidg over the universal coverM of M , and ∇ back to a Cartan connection∇ ong, as described in Section 2.6. Then, by Proposition 2.4, the finite-dimensional Lie algebra g 0 ⊂ Γ(g) of∇-parallel sections acts onM and this action is transitive. That this Lie algebra may be identified with the fibre g| m 0 , equipped with the bracket described in Proposition 5.1(1), follows from Remark 2.5 and Proposition 2.6(4) (but we make no further use of this interpretation). Identify Γ with the group of covering transformations and let us understand the monodromy representation φ → µ φ : Γ → Aut(g 0 ) in the invariant sense defined in Section 2.6. That Proposition 5.1(2) holds is just Proposition 2.6(2). According to Proposition 2.6(3), each transformation φ ∈ Γ is a g 0 -equivariant diffeomorphism of M with twist µ φ .
It is clear from Proposition 2.6(4) that geometric closure, in the sense of Section 1.2, is equivalent to the condition in Proposition 5.1(3) above. If ∇ is geometrically closed, we can define the development D :M → G 0 /H 0 associated with the action of g 0 onM , as determined by the choice of some fixed pointm 0 ∈M covering m 0 . Defining φ G 0 /H 0 as in (4.1), one obtains an action as described in Proposition 5.1(3) above, on account of Proposition 4.7(1) and (2).
The development D :M → G 0 /H 0 is a local diffeomorphism (Proposition 4.3) and, according to Proposition 4.7(3), transforms according to Now cover M with open sets U i , i ∈ I, small enough that each set U i is evenly covered by some open setŨ i ⊂M , and such that the restriction of the development D :Ũ i → G 0 /H 0 is a diffeomorphism onto its image. Refining this covering if necessary, we may arrange that each non-empty intersection U i ∩U j is connected (see, e.g., [6, Theorem I.5.1]). Let s i : U i →Ũ i denote the inverse of the restrictionŨ i → U i of the coveringM → M . Define charts ψ i : U i → G 0 /H 0 by ψ i = D • s i . Then, whenever U i ∩ U j = ∅, there exists a covering transformation φ ij ∈ Γ such that s j = φ ij • s i on U i ∩ U j . (Here one uses the fact that two local continuous sections of a covering map that have a common connected domain will agree on the entire domain if they agree at one point.) Furthermore, for any m ∈ U i ∩ U j , we have, by (5.1), Whence the maps ψ i : U i → G 0 /H 0 , i ∈ I, constitute an atlas of charts meeting the requirement in Proposition 5.1(4) above, and this concludes the proof of Proposition 5.1.

Completeness in terms of the associated Lie algebra action
Let g be a Lie algebroid on M (not necessarily transitive), let ∇ be a flat Cartan connection on g, and consider the associated action of the Lie algebra g 0 onM discussed in Section 2.6. Proof . Adopting the notation of Section 2.6, it is easy to see that ∇ is complete if and only if the pullback connection∇ ong is complete. But, according to Proposition 2.4(4),g is isomorphic to the action algebroid g 0 ×M , the connection∇ being represented by the canonical flat connection on g 0 ×M . But geodesics of the canonical flat connection on g 0 ×M are evidently those paths of the form X t = (ξ,m t ), where ξ ∈ g 0 andm t is an integral curve of the corresponding vector field ξ † .
Compactness of M is insufficient to guarantee completeness: Counterexample 6.2. Let M = S 1 be the circle and let g 0 = R act onM = R according to 1 † = e −θ ∂ ∂θ . Here θ denotes the standard coordinate function on R. Evidently, 1 † is not a complete vector field.
Define a representation µ : Γ → Aut(g 0 ) ∼ = R\{0} of the group of covering transformations Γ ∼ = Z on g 0 by µ n = e 2πn . As a covering transformation, each n ∈ Γ is the map θ → θ + 2πn, which is a g 0 -equivariant map with twist µ n . By the discussion at the end of Section 2.6, the quotient g = (M × g 0 )/Γ is a Lie algebroid over M = S 1 supporting a flat Cartan connection, but the corresponding Lie algebra action of g 0 onM is, by construction, incomplete.

Suf f icient conditions for completeness
Although compactness of M is not sufficient to guarantee completeness, compactness plus simple-connectivity is obviously sufficient, for thenM is also compact (and all vector fields on compact manifolds are complete). More generally, we have: Proposition 6.3. Let g be a Lie algebroid over M equipped with a flat Cartan connection ∇, and let µ : Γ → Aut(g 0 ) denote the associated monodromy representation, as described in Section 2.6. Assume that one of the following conditions holds: So, let m ∈ M be arbitrary. Then there is r = r(m) > 0 such that the geodesic ball B 3r (m) ⊂ M is well-defined, has compact closure, and is evenly covered by a disjoint union of geodesic balls B 3r (m) ⊂M , one for eachm lying over m in the covering. For any suchm we have, in the notation of the lemma, Proof that (2) implies completeness. In this case every vector field ξ † is a Killing field on the universal coverM , to which the metric σ lifts. In particular, every integral curve t → m(t) : (a, b) →M of ξ † has constant speed. Since M is a complete metric space, so isM .
Suppose that t n → b < ∞, for some sequence t 1 , t 2 , . . . ∈ (a, b). Because m(t) has constant speed, m(t 1 ), m(t 2 ), . . . is a Cauchy sequence, which must therefore converge to some m b ∈ M . Any integral curve of ξ † beginning at m b may, by uniqueness of integral curves, be regarded as an extension of m(t), i.e., m(t) extends to some time interval (a, b + ) with b + > b. A similar argument applies to a. Whence ξ † is complete by the usual proof by contradiction.

The proof of Theorem 1.2
With Proposition 6.1 in hand, the necessity of completeness in Theorem 1.2 is not difficult to prove, and we now turn to the proof of sufficiency. To that end, we begin with a variant of the result sought in which the model space G 0 /H 0 is the quotient of a simply-connected Lie group by a connected subgroup: Proposition 6.5. Let g be a transitive Lie algebroid, over a smooth connected manifold M , and ∇ a flat, geometrically closed, complete, Cartan connection on g. Then: (1) The subgroup H 0 ⊂ G 0 is closed, when G 0 and H 0 have the meanings given in Proposition 5.1, and the universal cover of M is G 0 /H 0 .
To recover Theorem 1.2 from this variant, i.e., to obtain M ∼ = (G/H)/Γ, with Γ ⊂ G acting by left translations, one defines G to be the semidirect product Γ ×μ G 0 as described already after the statement of Proposition 5.1.
Proof . Given a pair (g, ∇) satisfying the hypotheses of the proposition, consider the corresponding action of the Lie algebra g 0 on the universal coverM detailed in Section 2.6. In particular, G 0 is then the simply-connected Lie group integrating g 0 . According to Proposition 6.1 and Palais' integrability theorem [15], the action of g 0 can be integrated to an action of the Lie group G 0 . Therefore, in the definition of the development D :M → G 0 /H 0 of the g 0 -actionsee Section 4.2, but readM in place of M -we may take G to be the action groupoid G 0 ×M , and the groupoid morphism Ω : G → G 0 is just the projection Ω : G 0 ×M → G 0 . The subgroup H 0 ⊂ G 0 is closed because it is the isotropy subgroup of the G 0 -action atm 0 . Moreover, P = G 0 × {m 0 } and the development D :M → G 0 /H 0 is seen to be a diffeomorphism. This establishes (1).
In the subsequent identificationM ∼ = G 0 /H 0 , each covering transformation φ ∈ Γ and corresponding map φ G 0 /H 0 appearing in Proposition 5.1 is identified. The former therefore have the properties claimed in (2).

Illustrations
Following are simple illustrations of the theory expounded in the present article.

Local Lie groups
As in [3], we call pair of linear connections ∇,∇ on a smooth manifold M (more precisely, on its tangent bundle T M ) a dual pair if∇ X Y − ∇ Y X = [X, Y ]. Notice that a connection ∇ is a Cartan connection precisely when its dual∇ is flat. If both ∇ and∇ are flat, it follows from Theorem 1.1 that M is a locally homogeneous manifold modelled on some Lie group G. For the purposes of this section, we accordingly define a local Lie group to be a smooth connected manifold M , equipped with a dual pair of simultaneously flat connections ∇,∇.
A bone fide Lie group G 0 is a local Lie group in this sense: the canonical flat connections ∇,∇ -corresponding to right and left trivialization, respectively -furnish the required dual pair. In this case, both connections ∇ and∇ are complete (the geodesics being the right and left cosets of one-parameter subgroups of G) and have trivial monodromy. Conversely, we have: (1) The torsion, tor∇ = − tor ∇ of∇, is a∇-parallel tensor whose restriction to any tangent space g 0 = T m 0 M , m 0 ∈ M , is a Lie bracket.
If, additionally, ∇ is complete, and G 0 denotes the simply-connected Lie group integrating g 0 , then: (2) The universal cover of M is G 0 and the group of covering transformations Γ ∼ = π 1 (M ) acts by affine transformations.
If ∇ is both complete and has trivial monodromy, then: (3) There exists an embedding of Γ into G 0 such that M ∼ = G 0 /Γ.
Proof . The claim (1) is just a special case of Proposition 5.1(1). Conclusion (2) is a special case of Proposition 6.5. Under the additional hypothesis that ∇ has trivial monodromy, Proposition 6.5 (2) implies that each element of Γ ⊂ Aff(G 0 ) has trivial twist. Each such element is therefore a left translation, allowing us to identify Γ with a subgroup of G 0 , as claimed in (3).
Of course, in the case that the embedding Γ ⊂ G 0 in (3) is normal, M is a global Lie group. The problem of globalizing a local Lie group structure is also considered in [1] (under the additional assumption that the connection∇ comes from a global parallelism on M ). In particular, it is shown that a necessary condition for globalizability is the vanishing of the class [w] ∈ H 1 (M ), where w is the closed one-form defined by w(U ) = trace(tor∇(U, · )), which is closed on account of Bianchi's second identity and the hypothesis curv ∇ = 0.

Riemannian manifolds with constant scalar curvature
Let M be a smooth connected manifold. Associated with any Riemannian metric σ on M is a natural Lie subalgebroid g ⊂ J 1 (T M ) which, according to [4], supports a canonical Cartan connection ∇. The curvature of ∇ vanishes on any simply-connected region U precisely when the vector space of Killing fields on U is of maximal possible dimension, and the Lie algebra of all such Killing fields can be concretely described. We now give a global analogue of this result.
By definition, g is the subbundle of all one-jets J 1 m V of vector fields V on M such that σ has vanishing Lie derivative at the point m ∈ M . To describe the Cartan connection ∇ explicitly, let ∇ LC denote the Levi-Cevita connection, which we may regard as a splitting of the canonical exact sequence, There is a corresponding exact sequence, where h ⊂ T * M ⊗ T M denotes the o(n)-bundle of all σ-skew-symmetric tangent space endomorphisms, which is ∇ LC -invariant. Because σ is ∇ LC -invariant, the splitting subbundle of J 1 (T M ) lies inside g and we obtain an identification g ∼ = T M ⊕ h. Under this identification, the Cartan connection ∇ is given by With the help of the Bianchi identities for linear connections, one computes implying implying that ∇ is flat if and only if curv ∇ LC is both ∇ LC -invariant and h-invariant. Now h-invariance implies, by purely algebraic arguments, that curv ∇ LC has only a scalar component: for some function s ∈ R, the scalar curvature; ∇ LC -invariance implies s is a constant. With a view to applying Proposition 6.5 in the flat case, we first define g 0 and h 0 ⊂ g 0 as in Proposition 5.1, and, to obtain the bracket on g 0 , compute the torsion, tor∇, of the associated g-connection∇ on g (which is actually a g-representation). To this end, one requires a formula for the Lie bracket on g ⊂ J 1 (T M ). Indeed, under the identification g ∼ = T M ⊕ h, we have where [ · , · ] h is the bracket on the o(n)-bundle h. This formula is an instance of the general formula [4, Proposition 6.2(4)]. Supposing that ∇ is flat, and hence (7.1) holds, one obtains with the help of (7.2), Restricting tor∇ to obtain the Lie bracket on g 0 = g| m 0 , we see from the formula that g 0 is the Lie algebra of o(n) × R n (semidirect product), o(n + 1), or o(n, 1), according to whether s = 0, s > 0, or s < 0, respectively. In each case h 0 ⊂ g 0 is isomorphic to o(n).
By definition, the metric σ is g-invariant, so that completeness of M , as a metric space, implies completeness of the Cartan connection ∇, by Proposition 6.3. In that case, Proposition 6.5 establishes the following: (1) The universal cover of M is: • G 0 /H 0 ∼ = R n , in the case s = 0, where G 0 is the simply-connected Lie group with Lie algebra R n × o(n) (semidirect product); • G 0 /H 0 ∼ = S n , in the case s > 0, where G 0 is the simply-connected Lie group with Lie algebra o(n + 1); • G 0 /H 0 ∼ = H n , in case s < 0, where G 0 is the simply-connected Lie group with Lie algebra o(n, 1).
In every case H 0 ⊂ G 0 is the connected subgroup with Lie algebra o(n), realised as a subalgebra of g 0 in the usual way.