Cartan Connections on Lie Groupoids and their Integrability
A multiplicatively closed, horizontal n-plane field D on a Lie groupoid G over M generalizes to intransitive geometry the classical notion of a Cartan connection. The infinitesimalization of the connection D is a Cartan connection ∇ on the Lie algebroid of G, a notion already studied elsewhere by th...
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Blaom, A.D. 2019-02-18T15:14:02Z 2019-02-18T15:14:02Z 2016 Cartan Connections on Lie Groupoids and their Integrability / A.D. Blaom // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53C05; 58H05; 53C07 DOI:10.3842/SIGMA.2016.114 https://nasplib.isofts.kiev.ua/handle/123456789/148549 A multiplicatively closed, horizontal n-plane field D on a Lie groupoid G over M generalizes to intransitive geometry the classical notion of a Cartan connection. The infinitesimalization of the connection D is a Cartan connection ∇ on the Lie algebroid of G, a notion already studied elsewhere by the author. It is shown that ∇ may be regarded as infinitesimal parallel translation in the groupoid G along D. From this follows a proof that D defines a pseudoaction generating a pseudogroup of transformations on M precisely when the curvature of ∇ vanishes. A byproduct of this analysis is a detailed description of multiplication in the groupoid J¹G of one-jets of bisections of G. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Cartan Connections on Lie Groupoids and their Integrability Article published earlier |
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Cartan Connections on Lie Groupoids and their Integrability |
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Cartan Connections on Lie Groupoids and their Integrability Blaom, A.D. |
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Cartan Connections on Lie Groupoids and their Integrability |
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Cartan Connections on Lie Groupoids and their Integrability |
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Cartan Connections on Lie Groupoids and their Integrability |
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Cartan Connections on Lie Groupoids and their Integrability |
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cartan connections on lie groupoids and their integrability |
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Blaom, A.D. |
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Blaom, A.D. |
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A multiplicatively closed, horizontal n-plane field D on a Lie groupoid G over M generalizes to intransitive geometry the classical notion of a Cartan connection. The infinitesimalization of the connection D is a Cartan connection ∇ on the Lie algebroid of G, a notion already studied elsewhere by the author. It is shown that ∇ may be regarded as infinitesimal parallel translation in the groupoid G along D. From this follows a proof that D defines a pseudoaction generating a pseudogroup of transformations on M precisely when the curvature of ∇ vanishes. A byproduct of this analysis is a detailed description of multiplication in the groupoid J¹G of one-jets of bisections of G.
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1815-0659 |
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https://nasplib.isofts.kiev.ua/handle/123456789/148549 |
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Cartan Connections on Lie Groupoids and their Integrability / A.D. Blaom // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 27 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 114, 26 pages
Cartan Connections on Lie Groupoids
and their Integrability
Anthony D. BLAOM
10 Huruhi Road, Waiheke Island, New Zealand
E-mail: anthony.blaom@gmail.com
Received May 19, 2016, in final form December 02, 2016; Published online December 07, 2016
http://dx.doi.org/10.3842/SIGMA.2016.114
Abstract. A multiplicatively closed, horizontal n-plane field D on a Lie groupoid G
over M generalizes to intransitive geometry the classical notion of a Cartan connection.
The infinitesimalization of the connection D is a Cartan connection ∇ on the Lie algebroid
of G, a notion already studied elsewhere by the author. It is shown that ∇ may be regarded
as infinitesimal parallel translation in the groupoid G along D. From this follows a proof
that D defines a pseudoaction generating a pseudogroup of transformations on M precisely
when the curvature of ∇ vanishes. A byproduct of this analysis is a detailed description of
multiplication in the groupoid J1G of one-jets of bisections of G.
Key words: Cartan connection; Lie algebroid; Lie groupoid
2010 Mathematics Subject Classification: 53C05; 58H05; 53C07
1 Introduction
This article describes a generalization of classical Cartan connections using the language of
Lie groupoids, with which the reader is assumed to have some familiarity. We recommend the
introduction given in [7]. For more detail, see [12, 17, 23].
The present article provides some detail missing from our work published on related mat-
ters [3, 4, 5, 6] and is more technical than those works. Applications of the Lie groupoid approach
to Cartan connections will be given elsewhere.
1.1 Cartan connections on Lie groupoids
Let G be a Lie groupoid over a smooth connected manifold M and J1G the corresponding
Lie groupoid of one-jets of local bisections of G. Then a right-inverse S : G → J1G for the
canonical projection J1G→ G determines a n-plane field D ⊂ TG on G, where n = dimM . If
S : G→ J1G is additionally a morphism of Lie groupoids, we call D (or S) a Cartan connection
on G [4, 6]. In the general terminology of [16], S is a unital flat cleavage for the Lie groupoid
fibration J1G → G. Or, if we view the tangent bundle TG as a Lie groupoid over TM , then
D ⊂ TG should be multiplicatively closed in the sense of [14, 25].
Cartan connections on Lie groupoids are the global analogues of the Cartan connections on Lie
algebroids first introduced in [3] – here called infinitesimal Cartan connections – and studied
further in [4, 5]. Infinitesimal Cartan connections also appear in a new approach to Cartan
geometries introduced in the recent monograph [15] (where they are called Blaom connections).
Cartan connections in various incarnations have been central to the study of finite-type geo-
metric structures, and to the geometric approach to studying overdetermined systems of partial
differential equations. The first example can be found in Èlie Cartan’s seminal paper [9]. In the
present article a classical Cartan connection is a certain one-form on a principal H-bundle P
taking values in a vector space V with dimV = dimP , as first formulated by Ehresmann [18].
mailto:anthony.blaom@gmail.com
http://dx.doi.org/10.3842/SIGMA.2016.114
2 A.D. Blaom
See Section 2.1 for the definition and [26] for a nice introduction. Our first result, proven
in Sections 2 and 3.3, is a correspondence between classical Cartan connections and Cartan
connections on transitive Lie groupoids:
Theorem 1.1. Every classical Cartan connection ω on a principal H-bundle P determines
a canonical Cartan connection Sω on the corresponding gauge groupoid GP = (P × P )/H.
Conversely, every Cartan connection on a transitive Lie groupoid G coincides with Sω, for
some classical Cartan connection ω, after the identification G ∼= GP determined by a choice of
source-fibre P ⊂ G.
It should be noted here that our notion of a classical Cartan connection is, by default, stripped
of any ‘model data’ because such data plays no role at all in the above correspondence1.
It was shown already in [3] that every classical Cartan connection on a principal bundle P
determines an infinitesimal Cartan connection on the associated Atiyah Lie algebroid TP/H.
1.2 Examples
To illustrate the utility of Cartan connections in a simple example, suppose M is a Riemannian
manifold, and let M ×M be the pair groupoid. Regard elements of J1(M ×M) as one-jets of
local transformations of M and let G ⊂ J1(M ×M) be the subgroupoid of all one-jets J1
mφ for
which the tangent map Tmφ : TmM → Tφ(m)M is an isometry. Then it can be shown (see the
appendix) that G carries a canonical Cartan connection D whose significance is this: The first-
order extension b = J1φ of every local isometry φ of M is a local bisection of G integrating D
(i.e., D is tangent to b, when b is viewed as a submanifold of G). And, conversely, every n-
dimensional integral manifold of D is locally a local bisection of G arising as the extension of
a local isometry φ. In other words, the local isometries are in one-to-one correspondence with the
sufficiently localised n-dimensional integral manifolds of D. In particular, obstructions to the
involutivity (integrability) of D amount to obstructions to the existence of local isometries of M .
The main result of the present article, formulated in Section 1.4 below, concerns the infini-
tesimal (Lie algebroid) version of obstructions to the involutivity of a Cartan connection.
The prototype of an involutive Cartan connection D is the one associated with the action of
a Lie group G0 on M . In this case one takes G to be the action groupoid G0 ×M (transitive
only if the action is transitive) and defines a Cartan connection as follows: S(g,m) is the one-jet
at m of the constant bisection m′ 7→ (g,m′).
More examples of Cartan connections on Lie groupoids will be given elsewhere.
1.3 Infinitesimal Cartan connections
Infinitesimalizing a Cartan connection, viewed as a Lie groupoid morphism S : G → J1G, we
obtain a splitting s : g→ J1g of the exact sequence
0 −→ T ∗M ⊗ g −→ J1g −→ g −→ 0.
Here J1g denotes the Lie algebroid of one-jets of sections of the Lie algebroid g of G. (A natural
identification of the Lie algebroid of J1G with J1g is recalled in Section 3.1 below.) In the
category of vector bundles, splittings s of the above sequence are in one-to-one correspondence
with Koszul connections ∇ on g; this correspondence is given by
sX = J1X +∇X, X ∈ Γ(g).
(According to our sign conventions, the inclusion T ∗M ⊗ g→ J1g induces the following map on
sections: df⊗X 7→ fJ1X−J1(fX).) When s : g→ J1g is a morphism of Lie algebroids, as is the
1Model data is part of Ehresmann’s original formulation.
Cartan Connections on Lie Groupoids and their Integrability 3
case here, then ∇ is called a Cartan connection on g [3], or an infinitesimal Cartan connection
when we want to distinguish it from the corresponding global notion defined in Section 1.1. As
usual, the curvature of ∇ is defined by
curv∇(X,Y )Z = ∇X∇Y Z −∇Y∇XZ −∇[X,Y ]Z,
and ∇ is flat if curv∇ = 0.
1.4 The main theorem
The central question to be addressed in the present article is this:
To what extent are obstructions to the involutivity of a Cartan connection D encoded
in the corresponding infinitesimal Cartan connection ∇?
To answer this question, let D be a Cartan connection on a Lie groupoid G and let GD ⊂ G
denote the set of points through which there exist n-dimensional integral manifolds of D, n =
rankD = dimM .
Theorem 1.2.
(1) If a Cartan connection D on a Lie groupoid G over M is involutive in some neighborhood V
of M then the corresponding infinitesimal Cartan connection ∇ is flat.
(2) GD is a wide subgroupoid of G and, when ∇ is flat, a union of connected components of G
containing the source-connected component G0 of G.
The proof of this theorem and its connection to other work is discussed in Sections 1.6 and 1.7
below.
1.5 Pseudoactions
If ∇ above is flat then, replacing G with a union of connected components, D is tangent to
a foliation F on G. We now axiomatize the sense in which F is compatible with the multiplicative
structure of G, and how it accordingly generates a pseudogroup of local transformations on M
(the pseudogroup of isometries ofM in the case of a maximally symmetric Riemannian manifold).
Let G be a Lie groupoid and call an immersed submanifold Σ ⊂ G a pseudotransformation
if the restrictions to Σ of the groupoid’s source and target maps are local diffeomorphisms.
A pseudoaction [6] on G is any smooth foliation F on G such that:
(1) The leaves of F are pseudotransformations.
(2) F is multiplicatively closed.
To define what is meant in (2), regard local bisections of G as immersed submanifolds and let F̂
denote the collection of all local bisections that intersect each leaf of F in an open subset. Let Ĝ
denote the collection of all local bisections of G, this being a groupoid over the collection of all
open subsets of M . Then condition (2) is the requirement that F̂ ⊂ Ĝ be a subgroupoid.
Given a pseudoaction F of G on M , each element b ∈ F̂ defines a local transformation φb
of M . Let pseud(F) denote the set of all local transformations ψ : U → V of M that are locally
of this form; that is, for every m ∈ U , there exists a neighborhood U ′ ⊂ U of m such that
ψ|U ′ = φb for some for some b ∈ F̂ . Then, by (2), pseud(F) is a pseudogroup.
A pseudoaction generalizes the ordinary action of a Lie group G0 on M : Take G to be the
corresponding action groupoid G0 ×M and F the foliation whose leaves are {g} ×M , g ∈ G0.
For more examples see [6], where the following elementary observation is established:
4 A.D. Blaom
Proposition 1.3. An arbitrary foliation F on G is a pseudoaction if and only if its tangent
n-plane field D is a Cartan connection on G.
Under an additional completeness hypothesis, a pseudoaction becomes an ordinary Lie group
action, with possibly a monodromy ‘twist’ [6].
An immediate corollary to Theorem 1.2 can now be stated as follows:
Corollary 1.4. Assume G is source-connected. Then a Cartan connection D is tangent to
a pseudoaction F on G if and only if the corresponding infinitesimal Cartan connection ∇ is flat.
In practice, the pseudogroup generated by a pseudoaction F represents a pseudogroup of
symmetries, the Riemannian example mentioned in Section 1.2 being a case in point. We
consequently make the following interpretation:
A Cartan connection on a Lie groupoid is a (possibly intransitive) ‘symmetry de-
formed by curvature’.
1.6 Connections to other work
Theorem 1.2(1) and the above corollary already follow from a result of Salazar [25, Theo-
rem 6.4.1] (see also [14]) for arbitrary ‘multiplicatively closed’ distributions on a Lie groupoid, of
which the n-plane field D defined by a Cartan connection is a special case. The independently
obtained proof of Theorem 1.2 given in Section 5 is based on: (i) A detailed analysis of the
multiplicative structure of J1G, given in Section 3; and (ii) A characterization of the infinitesimal
connection ∇ as infinitesimal parallel translation in the Lie groupoid G along D, described in
Section 4. We expect both analyses will be of some independent interest.
In Salazar’s approach, one identifies the obstruction to the involutivity of D with a Lie
groupoid cocycle c, whose infinitesimalization κ, a Lie algebroid cocycle, is seen to vanish when∇
is flat. Van Est’s isomorphism between Lie groupoid and Lie algebroid cohomology, generalized
to Lie groupoids and algebroids [11], then implies that the cohomology class [c] must vanish. To
show c itself vanishes requires a further topological argument.
Although the basic idea behind Salazar’s proof and the present one are reasonably straight-
forward, flushing out the details requires some work in both cases. Salazar’s result has the
advantage that it applies in greater generality; we hope that our proof sheds light on the special
case important to the theory of Cartan connections.
As far as we know, the first statement (without proof) of Corollary 1.4 appears in [4, Ap-
pendix A].
By Corollary 1.4, the infinitesimalization of a pseudoaction is a flat Cartan connection on
a Lie algebroid, also called by us a twisted Lie algebra action, for reasons explained in [6].
Analogues of the so-called fundamental theorems, Lie I, Lie II and Lie III, for Lie groups and
Lie algebras, are established for pseudoactions and twisted Lie algebra actions in [6], to which
the present article can be regarded as a prequel.
The modern redevelopment of Cartan geometries along lines promoted here, and in our other
works cited above, has several aspects warranting further investigation. We briefly discuss two
here.
First, there have yet to appear many interesting applications of the theory to intransitive
geometric structures (structures that Ehresmann’s formulation cannot handle). One reason for
this may be the focus of current research on intransitive structures of infinite type, a subject
far richer than its transitive counterpart. These include: Dirac structures [10], which genera-
lize Poisson, presymplectic, and foliated structures; generalized complex structures [19]; and
Jacobi structures [20, 21, 22] which simultaneously generalize Poisson and contact structures
(see also [13]). However, add extra structure, or perturb the underlying ‘integrability condition’,
Cartan Connections on Lie Groupoids and their Integrability 5
and these structures generally become finite-type structures which presumably admit a Cartan
geometry interpretation. Moreover, adding extra structure, such as a compatible metric, may
shed light on the underlying infinite-type structure, in the same way that equipping a Riemann
surface with a metric sheds light on the underlying topology of the surface.
Secondly, quite a large number of examples of transitive geometric structures – including
conformal, CR, and projective structures – are of so-called parabolic type, a notion cast very
decidedly in Ehresmann’s formulation of a Cartan geometry (including model data) [8]. It would
be worthwhile finding an axiomatization of parabolic geometries in terms of Lie groupoids or Lie
algebroids. An intriguing observation in this direction is that the Atiyah algebroid of a parabolic
geometry is also a twisted Courant algebroid [1, 2, 27].
1.7 Multiplication in J1G and parallel translation
In making a proof of Theorem 1.2, we have been led to make a detailed investigation of the
multiplicative structure of J1G, which is given in Section 3. While not hard to guess at this
description – at the level of the bisection group, it is a semidirect product – in writing out details
the following technical problem arises: Given two paths in a Lie groupoid g(t), h(t) ∈ G such that
the product g(t)h(t) is always defined, compute the time derivative of the product g(t)h(t) in
terms of the time derivatives of the individual factors. In the category of Lie groups, a trivial ap-
plication of the chain rule solves the problem, but this application rests on the fact that g(t)h(s)
is defined for s 6= t, which needn’t be the case for arbitrary Lie groupoids. The solution adopted
here is to parallel translate the factors using an appropriate connection D on the groupoid.
As we show in Section 4, if one infinitesimalizes the parallel translation associated with a Car-
tan connection D on G, then one recovers the corresponding infinitesimal Cartan connection ∇
on g defined in Section 1.3 above. This alternative characterization of ∇ is the key to our proof
of Theorem 1.2.
2 Classical Cartan connections
We now construct, for each classical Cartan connection ω on a principal H-bundle P , a Cartan
connection Sω on the gauge groupoid (P×P )/H. A converse construction, completing the proof
of Theorem 1.1, is postponed until Section 3.3. In Section 2.4 we describe the infinitesimalization
of Sω, a Cartan connection ∇ω on the Atiyah Lie algebroid TP/H of P .
In this section only, G denotes a Lie group, and not a Lie groupoid.
2.1 Classical Cartan connections
Let H be a Lie group with Lie algebra h, let π : P → M be a right principal H-bundle, and
suppose P is equipped with a parallelism ω. Then ω is called a Cartan connection on P if:
(1) The infinitesimal generators ξ†, ξ ∈ h, of H are ω-parallel.
(2) The space of ω-parallel vector fields is invariant under the canonical right action of H on
the space of all vector fields (the action by pullback).
In this article, to distinguish this use of the term from the one given previously, we will call ω
a classical Cartan connection.
We may understand the parallelism ω as a one-form on P taking values in some vector
space V , and whose restriction to each tangent space is an isomorphism onto V . Since we
may identify V with the space of ω-parallel vector fields (the ones on which ω is constant)
condition (1) realizes h as a subpace of V and we obtain the tautology
(3) ω(ξ†(p)) = ξ, for all ξ ∈ h and p ∈ P .
6 A.D. Blaom
On the other hand, condition (2) means that V becomes a right representation of H, and this
action extends the adjoint action in the sense that ξh = Adh−1 ξ whenever ξ ∈ h. Tautologically,
then, we also have the equivariance condition
(4) ω(vh) = ω(v)h for all h ∈ H, v ∈ TP .
More commonly, one prescribes a vector space V , an inclusion h ⊂ V and a right representation
of H on V extending the adjoint action, and defines a Cartan connection as a parallelism
ω : TP → V satisfying (3) and (4). However, in this case it is easy to see that (3) and (4)
imply (1) and (2) and that consequently the two formulations are equivalent.
For example, if we take H to be a closed subgroup of some Lie group G, acting on G from
the right, then we may take V = g, where g is the Lie algebra of G, and take H to act on g by
the adjoint action: ξh := Adh−1 ξ, ξ ∈ g. Then the left-invariant Mauer–Cartan form ω on G is
a Cartan connection on the principal bundle G→ G/H satisfying the well-known Mauer–Cartan
equation dω − 1
2ω ∧ ω = 0, where a wedge implies a contraction using the bracket on g.2
2.2 Cartan connections with model data
Generalizing the previous example, we may consider Cartan connections on an arbitrary principal
H-bundle P →M for which V = g and H acts via restricted adjoint action. One refers to G/H
as the homogeneous model and the extra data (the Lie group G and the embedding H ⊂ G) as
model data. With model data prescribed, one defines the curvature of ω by Ω = dω − 1
2ω ∧ ω,
where the wedge again implies a contraction using the bracket on g (part of the model data!).
If Ω = 0, then M can be identified, at least locally, with the homogeneous model G/H [26,
Theorem 5.3]. More generally, this leads one to regard M with the given data as a homogeneous
space ‘deformed by curvature’.
Two points must be emphasised when comparing this ‘deformed by curvature’ interpretation
with the analogous interpretation of Cartan connections on Lie groupoids: First, the homoge-
neous model G/H is specified a priori; in particular, if M happens to be a different homogeneous
space, then generally Ω 6= 0. Secondly, only transitive symmetry can follow from the ‘flatness’
(Ω = 0) of a classical Cartan connection.
In fact, since every classical Cartan connection determines a Cartan connection on a Lie
groupoid, there is already a notion of curvature which does not depend on the prescription of
model data, but its interpretation is different: in the transitive case this curvature vanishes
when M is locally isomorphic to some homogeneous space, not necessarily a model fixed before-
hand. See also [5] and Remark 2.5.
The literature devoted to classical Cartan connections which include model data is large.
The reader is referred to the book [26] for an introduction, and to [8] for a more recent survey.
Note that an important work in which model data is not prescribed is Morimoto’s construction
of Cartan connections for filtered manifolds [24].
2.3 Cartan connections from classical Cartan connections
The notion of a parallelism on a smooth manifold P can be reformulated using Lie groupoid
language as follows: Let S : P ×P → J1(P ×P ) be any Cartan connection on the pair groupoid
P × P in the sense of Section 1.1. Note that elements of J1(P × P ) may be understood as one-
jets of local transformations φ : U → V of P . Now J1(P × P ), and hence the Lie subgroupoid
G = S(P × P ), acts canonically on the tangent bundle TP . Call a vector field on P parallel if
it is G-invariant, and denote the n-dimensional space of parallel vector fields by V, n = dimP .
2We define brackets on Lie algebras and Lie algebroids using right-invariant vector fields.
Cartan Connections on Lie Groupoids and their Integrability 7
The tautological map ω : TP → V satisfying ω(X(m)) = X for all X ∈ V is a parallelism, in
sense of Section 2.1, and evidently all parallelisms arise in this way.
Now suppose π : P →M is a right principal H-bundle and let H act on J1(P × P ) from the
right by Lie groupoid morphisms according to(
J1
pφ
)
h = J1
ph
(
Rh ◦ φ ◦Rh−1
)
,
where Rh(p) := ph, and let J1(P × P )h ⊂ J1(P × P ) denote the subset consisting of one-jets
of local transformations φ of P that are infinitesimally H-equivariant, which means φ∗ξ† = ξ†,
for all ξ in h. Then J1(P × P )h is a transitive H-invariant Lie subgroupoid of J1(P × P ), as is
readily verified.
Proposition 2.1. The parallelism ω : TP → V defined by a Cartan connection S on P × P is
a classical Cartan connection on P if and only if S satisfies the following conditions:
(1) S : P × P → J1(P × P ) takes values in J1(P × P )h.
(2) S(qh, ph) = S(p, q)h for p, q ∈ P and h ∈ H (H-equivariance).
The easy proof is left to the reader.
The key observation that allows one to drop a Cartan connection S on P × P to a Cartan
connection on the gauge groupoid (P × P )/H is the following:
Lemma 2.2. As Lie groupoids over M , we have J1((P × P )/H) ∼= J1(P × P )h/H.
The following comments will help the reader follow the proof:
(3) Our convention is to regard the source map of the pair groupoid P × P as (q, p) 7→ p.
(4) Given an element J1
pφ ∈ J1(P ×P )h represented by some infinitesimally H-equivariant φ,
the maps Rh ◦ φ and φ ◦Rh have the same germ at p, for all h ∈ H in some neighborhood
of the identity. This is because infinitesimal H-equivariance implies local H-equivariance.
(Consequently, for all H sufficiently close to the identity, we simply have (J1
pφ)h = J1
phφ.)
(5) Viewing a bisection of (P×P )/H as a right inverse for the source projection (q, p) modH 7→
π(p), the local triviality of the bundle π : P → M allows us to write such a bisection in
the form b(m) = (q(m), p(m)) mod H, for some smooth local right-inverse m 7→ p(m) of
π : P → M , and some smooth map m 7→ q(m), where the composite π ◦ q is a diffeomor-
phism onto its image.
Proof of Lemma 2.2. A map A : J1(P×P )h/H → J1((P×P )/H) is defined as follows. Given
an element J1
p0φ mod H ∈ J1(P × P )h/H represented by some infinitesimally H-equivariant
local transformation φ of P , first choose a local right-inverse m 7→ p(m) for π : P →M satisfying
p(m0) = p0. Then we put
A
(
J1
p0φ mod H
)
= J1
m0
b,
where m0 = π(p0) and b is the local bisection of (P × P )/H defined by
b(m) =
(
φ(p(m)), p(m)
)
mod H.
Let m 7→ p̄(m) be a second local right-inverse for π satisfying p̄(m0) = p0. Then p̄(m) =
p(m)h(m), for some smooth map m 7→ h(m) ∈ H satisfying h(m0) = id. For all m sufficiently
close to m0 we have φ(p(m)h(m)) = φ(p(m))h(m), by (4), in which case (φ(p̄(m)), p̄(m)) mod H
= (φ(p(m)), p(m)) mod H, which shows that the definition of b is (up to germ equivalence at m0)
independent of the choice of m 7→ p(m).
8 A.D. Blaom
To show that the definition of A(J1
p0φ mod H) is independent of the choice of representa-
tive J1
p0φ, it is enough to observe: (i) if J1
p̄0 φ̄ is a second representative then the one-jet of φ̄ at p̄0
coincides with the one-jet of Rh ◦ φ ◦Rh−1 , where h ∈ H is determined by p̄0 = p0h; and (ii) in
that case a local right-inverse m 7→ p̄(m) for π satisfying p̄(m0) = p̄0 is given by p̄(m) = p(m)h,
where p(m) is as above.
We leave it to the reader to show that A is a Lie groupoid morphism. To show A is injective,
let J1
m0
b mod H be an element of J1((P×P )/H) with b as in (5). Since each point of P in a neigh-
borhood of p0 := p(m0) is of the form p(m)h, for some uniquely determined h ∈ H and m ∈M ,
a local transformation φ of P is well-defined by φ(p(m)h) = q(m)h. Then φ is H-equivariant
by construction, and hence infinitesimally H-equivariant, and we have A(J1
p0φ mod H) = J1
m0
b.
The proof that A is injective is no more difficult and is omitted. �
Now let ω be a classical Cartan connection, understood as the parallelism defined by a Cartan
connection S on P × P satisfying (1) and (2). By the lemma, a Cartan connection Sω on the
quotient (P × P )/H is the same thing as a Lie groupoid morphism
Sω : (P × P )/H → J1(P × P )h/H
furnishing a right-inverse for the canonical projection J1(P × P )h/H → (P × P )/H. By the
preceding proposition, such a connection is well-defined by
Sω
(
(p, q) mod H
)
= S(p, q) mod H, p, q ∈ P.
2.4 Infinitesimalization of a classical Cartan connection
Let ω : P → V be a classical Cartan connection on a principal H-bundle as above. Then the
infinitesimalization of the Cartan connection Sω is a Cartan connection ∇ω on the Atiyah Lie
algebroid TP/H. To show how ∇ω is related to ω, let ∇̄ denote the infinitesimal parallelism
determined by ω, i.e., the flat Koszul connection on TP for which the notions of ω-parallel and
∇̄-parallel coincide; ∇̄ is characterised by the identity
ω(∇̄XY ) = d(ω(Y ))(X), X, Y ∈ Γ(TP ). (2.1)
Lemma 2.3. Let X and Y be H-invariant vector fields on P . Then:
(1) ∇̄XY is H-invariant.
(2) ∇̄XY = [X,Y ] whenever Y is vertical.
Proof. Since ω is H-equivariant, a vector field on P is H-invariant if and only if the V -valued
function ω(X) is H-equivariant. Given this, and using the fact that the time-t flow map of an
H-invariant vector field commutes with the action of H, it is not hard to prove (1). To prove (2)
it suffices, by (2.1), to prove that the Lie derivative LXω vanishes on vertical vectors. To this
end, let ξ ∈ h and p ∈ P be given. Then, denoting the time-t flow map of X by Φt
X , we compute
(LXω)(ξ†(p)) =
d
dt
ω
(
TΦt
X ·
d
ds
p exp(sξ)
∣∣∣
s=0
) ∣∣∣
t=0
=
d
dt
ω
(
d
ds
Φt
X(p exp(sξ))
∣∣∣
s=0
) ∣∣∣
t=0
=
d
dt
ω
(
d
ds
Φt
X(p) exp(sξ)
∣∣∣
s=0
) ∣∣∣
t=0
=
d
dt
ω
(
ξ†
(
Φt
X(p)
))∣∣∣
t=0
=
d
dt
ξ
∣∣∣
t=0
= 0.
The third equality holds because the time-t flow map of X commutes with the action of H. �
Cartan Connections on Lie Groupoids and their Integrability 9
Now the space of H-invariant vector fields on P is in one-to-one correspondence with the
sections of the Atiyah Lie algebroid TP/H. By part (1) of the lemma, the flat TP -connection ∇̄
therefore drops to a representation ∇̄ω of the Lie algebroid TP/H on itself. According to part (2)
of the lemma, we have, for sections X, Y of TP/H
∇̄ωXY = [X,Y ],
whenever #Y = 0. Here # denotes the anchor of TP/H and [·, ·] the canonical bracket on its
sections. It follows that there is a TM -connection ∇ω on TP/H defined implicitly by
∇̄ωXY = ∇ω#YX + [X,Y ].
The proof of the following is left as an exercise for the reader:
Proposition 2.4. The connection ∇ω is the infinitesimalization of the Cartan connection Sω.
In particular, ∇ω is an infinitesimal Cartan connection, a fact already observed in [3].
Remark 2.5. As a final remark, we observe that if we prescribe model data and define Ω =
dω − 1
2ω ∧ ω, then ∇ω is flat – implying Sω defines a pseudoaction on a union of connected
components of (P × P )/H – precisely when ∇̄Ω = 0 [3, Theorem C], a condition weaker than
the vanishing of Ω.
3 The multiplicative structure of J1G
Let G be a Lie groupoid. The Lie algebroid of J1G can be identified with J1g, where g denotes
the Lie algebroid of G (see Section 3.1 below). The canonical exact sequence,
0 −→ T ∗M ⊗ g −→ J1g −→ g −→ 0 (3.1)
may be regarded as the derivative of a natural sequence of groupoid morphisms,
Aut(TM, g) ↪→ J1G→ G.
Here Aut(TM, g) is a certain open neighborhood of the zero-section of T ∗M ⊗ g, defined in
Section 3.4 below; if G = M ×M , then Aut(TM, g) = Aut(TM).
Corresponding to (3.1) is an exact sequence of section spaces which splits canonically, leading
to an identification,
Γ
(
J1g
) ∼= Γ(g)⊕ Γ(T ∗M ⊗ g).
Under this identification, the Lie bracket on Γ(J1g) is a semidirect product [3].
In this section we establish the global analogue of this result, namely a semidirect product
structure,
B
(
J1G
) ∼= B(G)×B(Aut(TM, g)),
where B( · ) denotes the group of global bisections.
Just as a choice of Cartan connection ∇ on g determines an identification
J1g ∼= g⊕ (T ∗M ⊗ g)
and an associated semi-direct product structure for J1g, so a choice of Cartan connection on G
determines a semi-direct product structure for J1G. While we shall provide a direct demon-
stration of this fact, the reader may like to interpret the existence of the semi-direct product
10 A.D. Blaom
structure as a consequence of the fact that J1G → G is Lie groupoid fibration, and S a unital
flat cleavage for it [23, Theorem 2.5.3] (see also [16, Theorem 2.2.3]). In particular, S determines
a representation of G on the kernel J1
MG which defines the semi-direct product structure.
The non-trivial part of our proof amounts to finding an explicit formula for the representa-
tion of G on J1
MG using the adjoint representation of J1G on g (see Section 3.2 below) and our
concrete model Aut(TM, g) of J1
MG. This description rests on a detailed analysis of multipli-
cation in J1G. To formulate our claims in detail, which we do in Section 3.5, will require some
preparation.
3.1 Representing elements of J1G and its Lie algebra
There are three useful representations of an element of J1G. Formally, an element of J1G is
a one-jet at some m ∈M of a local bisection b : U → G of G. (It is our convention to regard local
bisections as locally defined right-inverses for the source projection α : G → M .) If g = b(m),
then the tangent map µ := Tmb is a linear map from TmM to TgG satisfying:
(1) Tmα ◦ µ = idTmM , and
(2) Tmβ ◦ µ : TmM → TmM is invertible.
Conversely, for any g ∈ G with α(g) = m, any linear map µ : TmM → TgG satisfying (1) and (2),
is the tangent map of some local bisection b : U → G whose one-jet at m ∈ M is independent
of the particular choice of b. We are therefore entitled to identity elements of J1G with linear
maps TmM → TgG with α(g) = m and satisfying (1) and (2), and will do so by default in the
sequel.
Finally, each µ : TmM → TgG as above may be identified with its image, a subspace of TgG
that is simultaneously a complement for the tangent space at g ∈ G to the fibre of the source
projection α : G → M through g, and the fibre of the target projection β : G → M through g;
and all such ‘simultaneous complements’ may be realised in this way.
Let g := L(G) denote the abstract Lie algebroid of G. Our convention is to regard an
element of g as a vector tangent at some m ∈ M ⊂ G to a fibre of the source projection
α : G→M . Let J1g denote the vector bundle of one-jets of sections of g. Then there is a natural
isomorphism θ : L(J1G)→ J1g whose inverse is described as follows: Let J1
mX ∈ J1g be given.
Then θ−1(J1
mX) will be a vector tangent to J1G at the identity element TmιM : TmM → TmM
of J1G (with ιM : M → G denoting the inclusion) and so may be viewed as an equivalence class
of paths in J1G passing through TmιM . Specifically,
θ−1
(
J1
mX
)
=
d
dt
Tm
(
Φt
XR ◦ ιM
)∣∣∣
t=0
, (3.2)
where XR denotes the corresponding right-invariant vector field on G and t 7→ Φt
XR its flow.
3.2 The adjoint representation
The adjoint representation of a Lie group G is a representation of G on its Lie algebra. More
generally, for each Lie groupoid G, one has a god-given representation of J1G on the Lie alge-
broid g of G, which we also call the adjoint representation; it is defined as follows: Let µ ∈ J1
gG
be given, where g ∈ G begins at m ∈ M and ends at m′ ∈ M , and let X ∈ g|m. First, notice
that we may use µ to lift X, viewed as a vector tangent to the source-fibre at m, to a vector X ′
tangent to the same source-fibre at g:
X ′ = µ(#X)− TLg · TI ·X.
Cartan Connections on Lie Groupoids and their Integrability 11
Here # denotes the anchor, I : G→ G inversion and Lg : G→ G left-multiplication by g. (One
has, incidentally, TI ·X = #X −X, for all X ∈ g.) Then we define
AdµX := TRg−1 ·X ′ = TRg−1 ·
(
µ(#X)− TLg · TI ·X
)
,
where Rg is right-multiplication by g. Notice that if #X = 0 (always true if G is a Lie group)
then this collapses to the familiar
AdµX = T
(
Rg−1 ◦ Lg
)
·X.
Additionally, there is a natural representation of J1G on TM , with respect to which the
anchor #: g→ TM is equivariant; this action, also denoted Ad, is given by
Adµ v := Tβ · µ(v), v ∈ TmM,
where β : G→M is the target projection.
Finally, the two representations just defined induce a representation of J1G on T ∗M ⊗ g
needed later:
(Adµ φ)v = Adµ
(
φAd−1
µ v
)
, φ ∈ (T ∗M ⊗ g)|m.
For the sake of completeness, we add the following observation:
Remark 3.1. The adjoint representation amounts to a Lie groupoid morphism
Ad: J1G→ GL(g),
where GL(g) is the frame groupoid of g. Infinitesimalizing, we obtain a Lie algebroid morphism
ad: J1g→ gl(g) coinciding with the adjoint representation of J1g on g, defined by
adJ1X Y = [X,Y ].
See, e.g., [3]. Here we are identifying the abstract Lie algebroid gl(g) of GL(g) with the Lie
algebroid of derivations on g according to the following convention: Given an element of the
abstract Lie algebroid represented by a path t 7→ φt ∈ GL(g) with φ0 the identity on g|m and φt
an isomorphism from g|m to g|mt , for some path t 7→ mt ∈M , the corresponding derivation ∂ is
defined by
∂Y =
d
dt
φ−1
t Y (mt)
∣∣∣
t=0
, Y ∈ Γ(g).
3.3 Recovering a classical Cartan connection
We pause our development to show how a Cartan connection S : G → J1G on a transitive Lie
groupoid G determines a classical Cartan connection ω on the source-fibre P over an arbitrary
fixed point m0 ∈M . Recall that P is a left principal H-bundle, where H is the group of arrows
of G simultaneously beginning at terminating at m0.
From the Cartan connection S we obtain a representation of G on its Lie algebroid g: gξ =
AdS(g) ξ. This is indeed a representation because S : G → J1G is a Lie groupoid morphism.
Letting V = g|m0 , we define ω : TP → V by ω(v) = g−1(TRg−1 · v), where g ∈ P is the base-
point of v ∈ TP . One readily verifies axioms (3) and (4) of Section 2.1 defining a classical
Cartan connection. Under the identification G ∼= (P ×P )/H the Cartan connection S coincides
with Sω, as defined in Section 2.3, as the reader is invited to check. This completes the proof
of Theorem 1.1.
12 A.D. Blaom
3.4 A linear model of the kernel of J1G→ G
Let J1
MG ⊂ J1G denote the pre-image of M ⊂ G under the natural projection J1G → G;
J1
MG is a totally intransitive Lie groupoid. We now define a certain open neighborhood of the
zero-section of T ∗M ⊗ g, denoted Aut(TM, g), that serves as a model of J1
MG.
For each φ ∈ T ∗M ⊗ g with base-point m ∈M , define φTM ∈ T ∗M ⊗ TM by
φTMv = v −#φv.
Then, writing φTM ∈ Aut(TM) if φTM : TmM → TmM is invertible, the set
Aut(TM, g) :=
{
φ ∈ T ∗M ⊗ g |φTM ∈ Aut(TM)
}
is an open neighborhood of the zero-section of T ∗M⊗g. Moreover, Aut(TM, g) becomes a totally
intransitive Lie groupoid over M , with the zero-section becoming the set of identity elements,
if we define multiplication by
ψφ = ψ + φ− ψ ◦# ◦ φ = φ+ ψ ◦ φTM , (3.3)
and inversion by
φ 7→ −φ ◦
(
φTM
)−1
.
With respect to this extra structure, the map
φ 7→ φTM : Aut(TM, g)→ Aut(TM)
is a groupoid morphism (in this case, just a fibre-wise group homomorphism) with commutative
kernel. In other words, for each m ∈ M , Aut(TM, g)|m is simply a commutative extension of
the image of φ 7→ φTM , a subgroup of Aut(TM)|m.
Finally, note that in addition to the natural representation of Aut(TM, g) on TM given by
φ · v = φTMv, we have representation of Aut(TM, g) on g given by
φ ·X = φgX := X − φ#X, X ∈ g.
With respect to these actions of Aut(TM, g) on g and TM , the anchor # is equivariant:
#φgX = φTM#X.
According to Theorem 3.3(1) below, we have Aut(TM, g) ∼= J1
MG as Lie groupoids.
3.5 Multiplication and inversion in J1G
Inversion in J1G is straight-forward to describe. We leave the proof of the following to the
reader:
Proposition 3.2 (inversion in J1G). Suppose g ∈ G begins at m ∈ M and ends at m′ ∈ M .
Then, for any µ ∈ J1
gG the inverse µ−1 ∈ J1
g−1G is given, as a map Tm′M → TgG, by
µ−1(v) = TI · µ
(
Ad−1
µ v
)
, v ∈ Tm′M.
Here I : G→ G denote inversion in G, and TI : TG→ TG its tangent map.
Cartan Connections on Lie Groupoids and their Integrability 13
We now state our results for multiplication. First, recall that the Lie groupoid Aut(TM, g)
is totally intransitive; its bisections coincide with its sections, as an open neighborhood of the
zero-section of the vector bundle T ∗M ⊗g. We have seen that the adjoint representation induces
an action of J1G on T ∗M ⊗ g. It is not hard to show that Aut(TM, g) ⊂ T ∗M ⊗ g is invariant
under this action, and we obtain an action of the bisection group B(G) on B(Aut(TM, g))
defined by
(b · Φ)(m′) = AdTmb Φ(m),
where β(b(m)) = m′ defines m ∈ M , and b ∈ B(G),Φ ∈ B(Aut(TM, g)),m ∈ M are arbitrary.
We may therefore form the semidirect product of groups,
B(G)×Ad B(Aut(TM, g)),
which has multiplication defined by
(b1,Φ1)(b2,Φ2) = (b1b2,Φ1b1 ·Φ2).
According to Theorem 3.3(2) below, this semidirect product is isomorphic to B(J1G).
Secondly, suppose G admits a Cartan connection S : G→ J1G. Then G acts on Aut(TM, g)
according to g ·φ := S(g)·φ = AdS(g) φ. With this action in hand, the pullback G×βAut(TM, g)
of the fibre-bundle Aut(TM, g) → M under the target projection β : G → M becomes a semi-
direct product of Lie groupoids: The source and target projections are respectively (g, φ) 7→ α(g)
and (g, φ) 7→ β(g), where α, β : G→ M are the corresponding projections for G; multiplication
is defined by
(g1, φ1)(g2, φ2) := (g1g2, φ1 AdS(g1) φ2).
According to Theorem 3.3(3) below, J1G ∼= G×β Aut(TM, g) as Lie groupoids.
Theorem 3.3. For an arbitrary Lie groupoid G:
(1) The map φ 7→ φ∨ : Aut(TM, g)→ J1G, defined by φ∨v = v − φv, is a Lie groupoid mor-
phism and embedding, whose image is the kernel of the natural projection J1G→ G. This
map is equivariant with respect to the natural representations of both groupoids on TM ,
and on g:
Adφ∨ v = φTMv, Adφ∨ X = φgX, v ∈ TM, X ∈ g.
(2) The map a : B(G)×Ad B(Aut(TM, g))→ B(J1G) defined by
a(b,Φ)(m) = (Φ(m′))∨Tmb, m′ = β(b(m)),
is an isomorphism of Lie groups.
Here Tmb : TmM → Tb(m)G is the tangent map of b at m, viewed as an element of J1G, and
β : G→M is the target projection. Also, (Φ(m′))∨Tmb denotes the product of (Φ(m′))∨ and Tmb
in the Lie groupoid J1G.
(3) If S : G → J1G is a Cartan connection on G, then the map c : G ×β Aut(TM, g) → J1G
defined by c(g, φ) = φ∨S(g) is an isomorphism of Lie groupoids.
14 A.D. Blaom
3.6 Source connections and parallel actions
Before proving the preceding results, we introduce some terminology and notation we need when
calculating the derivatives of paths in a groupoid G of the form g(t)h(t) (and later when we
establish a geometric interpretation of an infinitesimalized Cartan connection.)
By a source connection on G, we shall mean a rank-n subbundle D ⊂ TG such that: (i) TG =
D + kerTα; and (ii) D(m) = TmM for all m ∈ M ⊂ G. Here n := dim(M) and α : G → M
is the source projection. A Cartan connection D, as defined in Section 1.1 above, is a source
connection. While Cartan connections need not exist in general, one can always find a source
connection locally (which will suffice for present purposes) and even globally if we work in
the C∞ category.
Suppose a < 0 < b. Fixing a source connection D on G, and a smooth path γ : (a, b)→M on
the base M , we can define a local form of parallel translation along γ. This parallel translation
will be expressed in the language of Lie groupoid actions; see, e.g., [23, Section 1.6].
To simplify our description, we assume γ is regular and simple, so that the pullback γ∗G of
the surjective submersion α : G→M along γ : (a, b)→M can be identified with a submanifold
of G, namely the pre-image of γ((a, b)) under α. This pre-image is a union of source-fibres,
Pt = α−1(γ(t)), a < t < b. For any U ⊂ γ∗G we write (a, b)2 ×γ U for the set of all (t1, t2, g) ∈
(a, b)2 × U such that α(g) = γ(t1).
We claim that for any point g0 ∈ P0 ⊂ γ∗G (or, more generally, any finite number of points
in P0), there exists – shrinking the interval (a, b) if necessary – an open neighborhood U of g0
in γ∗G, and an action of the pair groupoid (a, b)2 on the restriction α : U → γ((a, b)), denoted
(t1, t2, g) 7→ Aγt1,t2(g) : (a, b)2×γ U → U , such that the path t 7→ Aγt0,t(g) is D-horizontal, for any
t0 ∈ (a, b) and g ∈ Pt0 . Indeed, t 7→ Aγt0,t(g) is then the D-horizontal lift through α : G → M ,
of the path t 7→ γ(t), that passes through g at time t0. The existence of the action follows, for
example, from the observation that the connection D on α : G → M pulls back to a rank-one
connection on the pullback γ∗G→ (a, b); finding horizontal paths then amounts to integrating
a vector field, to which the standard existence and uniqueness theory for flows applies.
We will call the action Aγ defined above the parallel action along γ associated with the
connection D. Note that Aγt1,t2(m) = m because D(m) = TmM for all m ∈M .
3.7 Multiplication in J1G in special cases
The technical part of the proof of Theorem 3.3 is to derive product formulas in a few special
cases. Referring to Fig. 1 below, we have:
Lemma 3.4. Suppose g ∈ G begins at m ∈ M and ends at m′ ∈ M . Then, for all µ, ν ∈ J1
gG
and φ ∈ Aut(TM, g)|m:
(1) µφ∨(v) = µ(φTMv) + TLg · TI · (φv), where v ∈ TmM is arbitrary;
(2) νµ−1 = ψ∨, where
ψv = TRg−1 ·
(
µ
(
Ad−1
µ v
)
− ν
(
Ad−1
µ v
))
, v ∈ Tm′M ;
(3) µφ∨µ−1 = (Adµ φ)∨.
Also, if µ, ν and φ above are related by ν = µφ∨, then:
(4) µ(v)− ν(v) = TRg ·Adµ(φv), v ∈ TmM .
Here TI, TLg, TRg−1 : TG→ TG denote the tangent maps for inversion, left-multiplication by g,
and right-multiplication by g−1. The map φ 7→ φ∨ was defined in Theorem 3.3(1).
Cartan Connections on Lie Groupoids and their Integrability 15
Figure 1. Schematic showing the first two products considered in Lemma 3.4. In the lemma we view
elements µ, ν, φ∨ ∈ J1G as linear maps but in the figure they are represented by their images (straight-line
segments).
Proof. We first prove (1). Let b1 and b2 be local bisections of G (viewed as local right inverses
of the source-projection G→M) such that
µ = Tmb1, φ∨ = Tmb2. (3.4)
Let v ∈ TmM be arbitrary and γ′ : (−ε, ε) → M be a simple regular path with γ′(0) = m′ and
γ̇′(0) = v, where a dot denotes derivative. According to the definition of multiplication in J1G,
we have
µφ∨(v) =
d
dt
g(t)h(t)
∣∣∣
t=0
, (3.5)
where
g(t) := b1(γ(t)), h(t) := b2(γ′(t)), γ(t) := β(h(t)) = β(b2(γ′(t))). (3.6)
Here β : G→M denotes the target projection. We note in passing that
γ̇(0) = φTMv. (3.7)
Unfortunately, we cannot apply the chain rule directly to (3.5) because g(s)h(t) need not
be defined unless s = t. To overcome this difficulty, equip G with an source connection D, as
described in Section 3.6 above, and let
(t1, t2, g) 7→ Aγt1,t2(g) : (−ε, ε)2 ×γ U → U
denote the associated parallel action along γ. Here ε > 0, and we may arrange U ⊂ G to be an
open set containing both elements m, g ∈ G. We may now write
g(t)h(t) = Aγ0,t(g̃(t))
(
Aγ0,t
(
h̃(t)−1
))−1
where
g̃(t) := Aγt,0(g(t)), (3.8)
h̃(t) :=
(
Aγt,0
(
h(t)−1
))−1
. (3.9)
16 A.D. Blaom
Noting that g(t)h(t) = f(t, t), where
f(s, t) = Aγ0,s(g̃(t))
(
Aγ0,s
(
h̃(t)−1
))−1
is well-defined for all (s, t) ∈ R2 sufficiently close to the origin, we may now apply the chain
rule to (3.5):
φ∨µ(v) =
d
ds
f(s, 0)
∣∣∣
s=0
+
d
dt
f(0, t)
∣∣∣
t=0
=
d
ds
Aγ0,s(g̃(0))
(
Aγ0,s
(
h̃(0)−1
))−1
∣∣∣
s=0
+
d
dt
Aγ0,0(g̃(t))
(
Aγ0,0
(
h̃(t)−1
))−1
∣∣∣
t=0
= w +
d
dt
g̃(t)h̃(t)
∣∣∣
t=0
, (3.10)
where w is the D-horizontal lift of γ̇(0) = φTMv to an element of D(g) ⊂ TgG.
Noting that g̃(s)h̃(t) is defined for all (s, t) ∈ R2 sufficiently close to the origin, we may apply
the chain rule directly to obtain
d
dt
g̃(t)h̃(t)
∣∣∣
t=0
= ˙̃g(0) + TLg
˙̃
h(0). (3.11)
It remains to compute ˙̃g(0) and
˙̃
h(0). From (3.8) we have
g(t) = Aγ0,t(g̃(t)) = Aγ0,s(g̃(t))
∣∣∣
s=t
,
so that the chain rule gives
ġ(0) =
d
ds
Aγ0,s(g̃(0))
∣∣∣
s=0
+
d
dt
Aγ0,0(g̃(t))
∣∣∣
t=0
= w + ˙̃g(0).
Therefore
˙̃g(0) = ġ(0)− w.
Similar arguments applied to (3.9) give us
˙̃
h(0) = ḣ(0)− φTMv.
These last two results and (3.11) imply
d
dt
g̃(t)h̃(t)
∣∣∣
t=0
= ġ(0)− w + TLg ·
(
ḣ(0)− φTMv
)
.
Equation (3.10) now reads
φ∨µ(v) = ġ(0) + TLg ·
(
ḣ(0)− φTMv
)
.
Finally, appealing to (3.6), (3.7) and (3.4), we obtain
φ∨µ(v) = µ
(
φTMv
)
+ TLg ·
(
φ∨(v)− φTMv
)
.
Noting the identity #X −X = TI ·X, X ∈ g, one obtains
TLg ·
(
φ∨(v)− φTMv
)
= TLg · (#φv − φv) = TLg · TI · (φv).
Formula (1) now follows.
Cartan Connections on Lie Groupoids and their Integrability 17
The strategy for proving (2) is very similar and is omitted. Regarding (3), we have, putting
ν = µφ∨ in (2), µφ∨µ−1 = ψ∨, where
ψv = TRg−1 ·
(
µ(w)− (µφ∨)(w)
)
,
and w = Ad−1
µ v. Substituting (1):
ψv = TRg−1 ·
(
µ(w)− µ(φTMw)− TLg · TI · (φw)
)
= TRg−1 · (µ(#φw)− TLg · TI · (φw))
= Adµ(φw) = Adµ
(
φAd−1
µ v
)
= (Adµ φ)v.
Finally, we note that (4) is a straightforward consequence of (3) and (2). �
3.8 Proof of Theorem 3.3
That the map φ 7→ φ∨ in Theorem 3.3(1) is an embedding, with image the kernel of J1G→ G,
is readily checked, as are the equivariance claims. To prove ψ∨φ∨ = (ψφ)∨, take g = m and
µ = ψ∨ in Lemma 3.4(1) and apply the identity TI · X = #X − X, for X ∈ g. Note that
multiplication in Aut(TM, g) was defined in (3.3).
To prove Theorem 3.3(2), let b1, b2 be bisections of G and Φ1, Φ2 sections of Aut(TM, g).
Let m ∈M be given and define m′ = β(b2(m)) and m′′ = β(b1(m′)). We compute
a
(
(b1,Φ1)(b2,Φ2)
)
(m) = a
(
b1b2,Φ1 b1 · Φ2
)
(m) =
(
(Φ1 b1 · Φ2)(m′′)
)∨
Tm(b1b2)
=
(
Φ1(m′′) AdTm′b1 Φ2(m′)
)∨
Tm(b1b2)
by Theorem 3.3(1)
=
(
Φ1(m′′)
)∨(
AdTm′b1 Φ2(m′)
)∨
Tm(b1b2)
by Lemma 3.4(3)
=
(
Φ1(m′′)
)∨
Tm′b1
(
Φ2(m′)
)∨
(Tm′b1)−1Tm(b1b2)
=
(
Φ1(m′′)
)∨
Tm′b1
(
Φ2(m′)
)∨
Tmb2 = a(b1,Φ1)a(b2,Φ2).
Regarding, Theorem 3.3(3), the reader will be readily convinced that the map
c : G×β Aut(TM, g)→ J1G
defined by c(g, φ) = φ∨S(g) is a diffeomorphism. To show that it is a morphism of groupoids,
let (g1, φ1), (g2, φ2) ∈ G×β Aut(TM, g) be given. Then
c(g1, φ1)c(g2, φ2) = φ∨1S(g1)φ∨2S(g2) = φ∨1S(g1)φ∨2S(g1)−1S(g1g2)
by Lemma 3.4(3)
= φ∨1 (AdS(g1) φ2)∨S(g1g2)
by Theorem 3.3(1)
= (φ1 AdS(g1) φ2)∨S(g1g2)
= c(g1g2, φ1 AdS(g1) φ2) = c
(
(g1, φ1)(g2, φ2)
)
.
4 On the infinitesimalization of Cartan connections
Let D ⊂ TG be a Cartan connection on a Lie groupoid G over M . Then, in the terminology
of Section 3.6, D is also a globally defined source-connection on G. We differentiate the cor-
responding parallel action of D, along curves in M , to obtain a linear connection ∇D on the
Lie algebroid g of G. Theorem 4.1 below states that this ‘geometrically’ defined connection ∇D
coincides with the infinitesimal Cartan connection ∇, defined ‘algebraically’ in Section 1.3.
18 A.D. Blaom
4.1 Differentiating the parallel action defined by a source connection
Let D be an arbitrary source connection on G. Let v ∈ TmM be an arbitrary tangent vector at
some point m ∈ M , and γ : (−ε, ε) → M a simple regular path with γ(0) = m and γ̇(0) = v,
where a dot denotes derivative. Shrinking ε if necessary, we have, for some open neighborhood U
of m in γ∗G, the corresponding parallel action Aγ : (−ε, ε)2×γU → U defined in Section 3.6 – an
action of the pair groupoid (−ε, ε)2 on U . We obtain an action aγ of (−ε, ε)2 on the pullback γ∗g –
which, for simplicity, we identify with a submanifold of g – by differentiating:
aγt1,t2
(
d
ds
g(s)
∣∣∣
s=0
)
:=
d
ds
Aγt1,t2g(s)
∣∣∣
s=0
.
Here we view an element of γ∗g|m ∼= g|m as the derivative of some path s 7→ g(s) ∈ G lying in
a source-fibre of G. A linear connection ∇D on g is now defined by
∇Dv X := ∂ta
γ
t,0X(γ(t))|t=0, X ∈ Γ(g).
Notation. Here and in the sequel ∂t indicates a derivative with respect to t of a vector-valued
function of t, identified with an element of the underlying vector space and not a tangent vector
(i.e., with base-point ‘forgotten’).
Here is the main result of the present section:
Theorem 4.1. If D ⊂ TG is a Cartan connection on G then ∇D coincides with the correspon-
ding infinitesimal Cartan connection ∇ defined in Section 1.3.
The proof of this theorem is given in Section 4.3 after necessary preparations.
4.2 The Lie algebroid of Aut(TM, g)
Implicit in the discussion at the beginning of Section 3 is an identification of the Lie algebroid of
our model Aut(TM, g) of the kernel of J1G→ G with the kernel T ∗M ⊗ g of J1g→ g. To prove
the preceding theorem we must make this identification explicit. To this end, let L(G) denote
the (abstract) Lie algebroid of any Lie groupoid G. Then an element of L(Aut(TM, g)) is of
the form d
dtφt|t=0, for some path t 7→ φt ∈ Aut(TM, g) lying completely in some source-fibre
(= target-fibre) of Aut(TM, g); the latter is simply an open neighborhood of zero in a vector
space
(T ∗M ⊗ g)|m, m ∈M.
Of course, we also require that φ0 be an identity element of Aut(TM, g), which just means
that φ0 is the zero element of (T ∗M ⊗ g)|m. There is a natural isomorphism of vector bundles
θ′ : L(Aut(TM, g))→ T ∗M ⊗ g defined by
θ′
(
d
dt
φt
∣∣∣
t=0
)
v = ∂tφtv|t=0.
This isomorphism is the appropriate one for present purposes, by virtue of the following:
Proposition 4.2. Defining θ′ as above, we have the following commutative diagram with exact
rows:
0 −−−−→ L(Aut(TM, g))
j′−−−−→ L(J1G) −−−−→ L(G) −−−−→ 0
θ′
y θ
y ∥∥∥
0 −−−−→ T ∗M ⊗ g
j−−−−→ J1g −−−−→ g −−−−→ 0.
Cartan Connections on Lie Groupoids and their Integrability 19
Here θ denotes the isomorphism described in Section 3.1; j′ denotes the derivative of the map
φ 7→ φ∨ : Aut(TM, g) → J1G defined in Theorem 3.3(1); j denotes the map which induces
a corresponding map of section spaces given by df ⊗X 7→ fJ1X − J1(fX).
Key in the proof of the proposition is the following technical result applied again in Section 4.3
below:
Lemma 4.3. Let α : G → M denote the source map of G and let D be any source-connection.
For each g ∈ G beginning at some m = α(g), let S(g) : TmM → D(g) denote the inverse of
the restriction Tgα : D(g) → TmM of the tangent map Tα. (In other words, S(g)(v) is the
D-horizontal lift of v ∈ TmM to a vector at g; if D is a Cartan connection, we view S(g) ∈ J1G
and S : G → J1G has the meaning in Section 1.1.) Then, for any function f on G defined in
a neighborhood of m, one has〈
df,∇Dv X
〉
= ∂t
〈
df, Tm
(
Φt
XR ◦ ιM
)
· v
〉∣∣
t=0
− ∂t
〈
df, S
(
Φt
XR(m)
)
(v)
〉∣∣
t=0
.
Proof. Adopting the notation of Section 4.1 with v = γ̇(0), we calculate,〈
df,∇Dv X
〉
= ∂s
〈
df, aγs,0X(γ(s))
〉∣∣
s=0
= ∂s∂tf
(
Aγs,0Φt
XR(γ(s))
)∣∣
t=0,s=0
= ∂t∂sf(g(s, t))|t=0,s=0,
i.e.,
〈
df,∇Dv X
〉
= ∂t
〈
df,
∂
∂s
g(s, t)
∣∣∣
s=0
〉 ∣∣∣
t=0
, (4.1)
where g(s, t) = Aγs,0Φt
XR(γ(s)) is defined for all (s, t) sufficiently close to (0, 0), and we have
appealed to the equality of mixed partial derivatives. Now we can write Φt
XR(γ(s)) = Aγ0,sg(s, t)
and notice that Aγ0,s1g(s2, t) is defined for all (s1, s2, t) sufficiently close to (0, 0, 0). It follows
that we may differentiate with respect to s at s = 0 and apply the chain rule:
∂
∂s
Φt
XR(γ(s))
∣∣∣
s=0
=
∂
∂s
Aγ0,sg(0, t)
∣∣∣
s=0
+
∂
∂s
Aγ0,0g(s, t)
∣∣∣
s=0
,
i.e.,
Tm
(
Φt
XR ◦ ιM
)
· v = S(Φt
XR)(v) +
∂
∂s
g(s, t)
∣∣∣
s=0
.
We conclude that
∂
∂s
g(s, t)
∣∣∣
s=0
= Tm
(
Φt
XR ◦ ιM
)
· v − S
(
Φt
XR(m)
)
(v).
Substituting this into (4.1) gives the desired result. �
Proof of Proposition 4.2. Exactness of the top row follows from Theorem 3.3(1); exactness
of the bottom row is immediate. It remains to show that for any ζ ∈ L(Aut(TM, g)), we have
θ(j′(ζ)) = j(θ′(ζ)). (4.2)
By exactness, and the fact that θ is an isomorphism, the element j′(ζ) is in the image of the
composite morphism
T ∗M ⊗ g
j−→ J1g
θ−1
−−→ L
(
J1G
)
.
20 A.D. Blaom
But any element of T ∗M ⊗ g is of the form −∇X(m), where ∇ is an arbitrary linear connection
on g, m ∈ M , and X is a local section of g with X(m) = 0. Since in that case j(−∇X(m)) =
J1
mX, it follows, from formula (3.2) for θ−1 that
j′(ζ) =
d
dt
Tm
(
Φt
XR ◦ ιM
)∣∣∣
t=0
, (4.3)
for some m ∈ M and X ∈ Γ(g) with X(m) = 0. Furthermore, we have, by construction,
θ(j′(ζ)) = j(−∇X(m)). Consequently, to prove (4.2), it suffices, since j is injective, to show
that
∇X(m) = −θ′(ζ). (4.4)
Now the map j′ is the derivative of the embedding φ 7→ φ∨ : Aut(TM, g)→ J1G defined in
Theorem 3.3(1). In view of this fact and (4.3), there is a path t 7→ φt in Aut(TM, g) such that
φ∨t = Tm(Φt
XR ◦ ιM ) and ζ = d
dtφt|t=0. To prove (4.4) we will apply both sides to some tangent
vector v = γ̇(0) ∈ TmM and test the resulting elements of g|m on a test function f , defined
on G, in some neighborhood of m. It will be convenient to take ∇ = ∇D, where D ⊂ TG is an
arbitrary source connection on G. Then, applying the preceding lemma, and using the fact that
Φt
XR(m) = m (since X(m) = 0) we compute
〈df,∇vX〉 = 〈df,∇Dv X〉 = ∂t
〈
df, Tm(Φt
XR ◦ ιM ) · v
〉∣∣
t=0
− ∂t〈df, v〉|t=0
= ∂t
〈
df, φ∨t (v)− v
〉∣∣
t=0
= ∂t
〈
df,−φtv
〉∣∣
t=0
= −〈df, ∂t(φtv)〉|t=0 = −
〈
df, θ′
(
d
dt
φt
∣∣∣
t=0
)
v
〉
= −〈df, θ′(ζ)v〉,
which proves (4.4). �
4.3 The proof of Theorem 4.1
Lemma 4.4. Let gt be a path in G such that g0 = m ∈M and gt begins at m for all t. Suppose
µt and νt are two paths in J1G with common projection gt onto G, and such that µ0 and ν0
coincide with the identity element TmιM of J1G (see Fig. 2). Then there exists a (necessarily
unique) ψ ∈ T ∗M ⊗ g such that:
(1) θ
(
d
dtµt
∣∣
t=0
− d
dtνt
∣∣
t=0
)
= j(ψ), and
(2) 〈df, ψv〉 = −∂t〈df, µt(v)〉|t=0 + ∂t〈df, νt(v)〉|t=0,
for all functions f on G defined in some neighborhood of m, and all v ∈ TmM .
For the definition of θ and j see Proposition 4.2.
Proof of Lemma 4.4. Since µt and νt have common projection onto G, we have νt = φ∨t µt
(multiplication in J1G) for some φt ∈ T ∗M⊗g. We have φ∨0 = TmιM , i.e., φ0 = 0. Differentiating
and applying the chain rule, we obtain
d
dt
µt
∣∣∣
t=0
− d
dt
νt
∣∣∣
t=0
= − d
dt
φ∨t
∣∣∣
t=0
= −j′
(
d
dt
φt
∣∣∣
t=0
)
,
where j′ is defined in Proposition 4.2. Applying θ to both sides, and using commutativity of the
diagram appearing in that proposition, we obtain (1) if we take ψ = −θ′( ddtφt). In that case, by
the definition of θ′, we have
ψv = −∂t(φtv)|t=0, v ∈ TmM. (4.5)
Cartan Connections on Lie Groupoids and their Integrability 21
Figure 2. Schematic showing the paths in J1G considered in Lemma 4.4.
On the other hand, applying Lemma 3.4(4), we have
TRgt ·Adµt(φtv) = µt(v)− νt(v).
Applying df to both sides and differentiating using the chain rule on the left, we obtain
∂t〈df, TRgt ·Adµt ζTG(m)〉|t=0 + ∂t〈df, φtv〉|t=0 = ∂t〈df, µt(v)〉|t=0 − ∂t〈df, νt(v)〉|t=0.
Here ζTG denotes the zero-section of TG. We have TRgt ·Adµt ζTG(m) = ζTG(gt), which means
the first term on the left vanishes, giving us
〈df, ∂t(φtv)|t=0〉 = ∂t〈df, µt(v)〉|t=0 − ∂t〈df, νt(v)〉|t=0.
This, together with (4.5), establishes (2). �
Now suppose D ⊂ TG is a Cartan connection on G, let S : G → J1G denote the corre-
sponding morphism of Lie groupoids, and ∇ the corresponding infinitesimal Cartan connection
on g = L(G). We are now in a position to prove ∇D = ∇.
First, let us make the ‘algebraic’ definition of ∇ in Section 1.3 more explicit. If we denote
the (abstract) derivative of S : G → J1G by dS : L(G) → L(J1G) then we obtain a splitting
s : g→ J1g of the exact sequence appearing in the bottom row of the commutative diagram in
Proposition 4.2, by putting s = θ ◦ dS. Then, ∇ is defined implicitly by
j(∇X(m)) = −J1
mX + s(X(m)).
Recalling the definition of θ given in (3.2), we may rewrite this as
j(∇X(m)) = θ
(
− d
dt
Tm
(
Φt
XR ◦ ιM
)∣∣∣
t=0
+
d
dt
S
(
Φt
XR(m)
)∣∣∣
t=0
)
.
Applying the preceding lemma, we obtain
〈df,∇vX〉 = ∂t
〈
df, Tm
(
Φt
XR ◦ ιM
)
· v
〉
|t=0 − ∂t
〈
df, S
(
Φt
XR(m)
)
(v)
〉∣∣
t=0
.
Here v ∈ TmM is arbitrary, as is the locally defined function f on G. Our claim ∇ = ∇D now
follows from Lemma 4.3.
5 Proof of Theorem 1.2
In this section D is a Cartan connection on G, S : G→ J1G the corresponding right-inverse for
the natural projection J1G→ G, and ∇ the corresponding infinitesimal Cartan connection.
22 A.D. Blaom
5.1 Proof of Theorem 1.2(1)
Let F be the foliation on V ⊂ G integrating D. For each m ∈M let Pm ⊂ V be the intersection
of V with the source fibre of G over m. Then Pm is transverse to F . In some open neighborhood
U ⊂ G of m0, the leaves of F intersect U in level sets of a submersion Ω: U → Pm0 . Now let
v = γ̇(0) ∈ TmM be the derivative of some path γ on M , m ∈ M ∩ U . Then, denoting by Aγ
the parallel action along γ determined by D, we have
Ω
(
Aγt,0(g)
)
= Ω(g), (5.1)
for all t sufficiently close to zero, and all g ∈ Pγ(t) sufficiently close to γ(t) ∈M .
If g|M∩U denotes the restriction of g, and g0 := Tm0Pm0 , then we obtain a map ω : gM∩U → g0
by differentiating Ω: U → Pm0 :
ω
(
d
ds
g(s)
∣∣∣
s=0
)
:=
d
ds
Ω
(
g(s)
)∣∣∣
s=0
. (5.2)
As usual, we are viewing an element of g as an equivalence class of paths in a source fibre.
Applying Theorem 4.1 we calculate
ω(∇vX) = ω
(
∂t
∂
∂s
Aγt,0
(
Φs
XR(γ(t))
)∣∣
s=0,t=0
)
by (5.2)
= ∂t
∂
∂s
Ω
(
Aγt,0
(
Φs
XR(γ(t))
))∣∣∣
s=0,t=0
by (5.1)
= ∂t
∂
∂s
Ω
(
Φs
XR(γ(t))
)∣∣∣
s=0,t=0
= ∂tω
(
X(γ(t))
)
|t=0 = 〈d(ω(X)), v〉.
So ω(∇VX) = LV (ω(X)), for any vector field V on M ∩U , where L denotes Lie derivative. For
any two vector fields V1 and V2 on M ∩ U we deduce
ω
(
curv∇(V1, V2)X
)
= LV1LV2(ω(X))− LV2LV1(ω(X))− L[V1,V2](ω(X)) = 0.
This shows that ∇ is flat in a neighborhood of the arbitrarily fixed point m0 ∈M .
The remaining sections are devoted to the proof of part (2) of Theorem 1.2.
5.2 Proof that GD ⊂ G is a subgroupoid
This follows immediately from the following lemma (used again in Section 5.4 below):
Lemma 5.1. Let b1 : U1 → G and b2 : U2 → G be two multipliable local bisections integrating D.
Then their product b1b2 integrates D.
Proof. Let m be an arbitrary point in U2 (the domain of the product b1b2) and let g1 = b1(m)
and g2 = b2(m). The arrow g2 terminates at some point m′ in the domain of b1. Since b1
and b2 integrate D, the one-jet S(g2) of a local bisection at m has representative b2, while
the one-jet S(g1) of a local bisection at m′ has representative b1. By the definition of the
product in J1G, S(g1)S(g2) is the one-jet at m of the product b1b2, a one-jet that must coincide
with S(g1g2), because D is a Cartan connection. So the image of the tangent map of b1b2 at m
must be D(g1g2). Since m was arbitrary, this shows b1b2 integrates D. �
Since the identity bisection lies in GD, GD is wide in G.
From now on, we suppose that ∇ is flat.
Cartan Connections on Lie Groupoids and their Integrability 23
5.3 Proof that GD contains an open neighborhood of M
Our proof that GD ⊂ G is open will rest on the fact that each point m ∈ M has an open
neighborhood lying in GD. In fact, we now prove the following stronger result:
Proposition 5.2. Assume M is simply-connected and that G has simply-connected source-fibres.
Then, assuming ∇ is flat, GD = G.
We begin by recalling that every Lie algebroid equipped with a flat Cartan connection ∇ is
an action algebroid, if M is simply-connected [3] (for generalizations, see [5]):
Lemma 5.3. Let g be a Lie algebroid over a simply-connected manifold M supporting a flat
Cartan connection ∇. Then the subspace g0 ⊂ Γ(g) of ∇-parallel sections is a Lie subalgebra
acting on M , with the corresponding Lie algebra homomorphism ξ 7→ ξ† : g0 → Γ(TM) being
given by ξ†(m) := #ξ(m). Moreover, the vector bundle morphism (ξ,m) → ξ(m) : g0 ×M → g
is a Lie algebroid isomorphism between the action algebroid g0 ×M and g.
Proof of Proposition 5.2. Applying the lemma to the Lie algebra g of G, we obtain a Lie
algebra g0 and a morphism of Lie algebroids ω : g→ g0, such that ω restricted to any fibre g|m
is an isomorphism onto g0. Since G is source-simply-connected, there exists a Lie groupoid
morphism Ω: G→ G0 integrating ω : g→ g0, by Lie II for Lie groupoids (see, e.g., [12]). Here G0
is the simply-connected Lie group integrating g0. Because ω is a point-wise isomorphism, the
restriction of Ω to any source-fibre of G, or any target-fibre of G, is a local diffeomorphism.
In particular, each level set Ω−1(g0) of Ω is a pseudotransformation in the sense of Section 1.5.
Moreover, as Ω is a morphism of Lie groupoids, it is not hard to see that the foliation of G by
connected components of these level sets is a pseudoaction of G. The tangent n-plane field D′ is
accordingly a Cartan connection on G, by Proposition 1.3. Since D′ is integrable by construction,
to prove the proposition, it suffices to show D = D′.
Now D = D′ precisely when the corresponding Lie groupoid morphisms
S, S′ : G→ J1G
are the same. Since G has connected source-fibres, it actually suffices to show that the deriva-
tives s, s′ : g → J1g coincide, or equivalently, that the corresponding infinitesimal Cartan con-
nections ∇, ∇′ are the same.
Now∇ is flat by hypothesis, and∇′ is flat by Theorem 1.2(1), already proven above. Since M
is simply-connected, to show ∇′ = ∇ it is therefore sufficient to show that ∇-parallel sections
of g are also ∇′-parallel. Now X ∈ Γ(g) is ∇-parallel precisely when ω(X) is a constant ξ. Let
v = γ̇(0) ∈ TmM be the derivative of some path γ on M . Also, let Aγ denote the parallel action
along γ defined by the connection D′. Then, by the definition of D′, we have
Ω
(
Aγt,0(g)
)
= Ω(g), (5.3)
for all t sufficiently close to 0 and all g ∈ α−1(γ(t)) sufficiently close to γ(t) ∈M . On the other
hand, applying Theorem 4.1 to the connection ∇′, we have
ω(∇′vX) = ω
(
∂t
∂
∂s
Aγt,0
(
Φs
XR(γ(t))
)∣∣
s=0, t=0
)
= ∂t
∂
∂s
Ω
(
Aγt,0
(
Φs
XR(γ(t))
))∣∣∣
s=0,t=0
by (5.3)
= ∂t
∂
∂s
Ω
(
Φs
XR(γ(t))
)∣∣∣
s=0, t=0
= ∂tω
(
X(γ(t))
)
|t=0 = ∂tξ|t=0 = 0. �
24 A.D. Blaom
5.4 Proof that GD ⊂ G is open
To show GD ⊂ G is open, let g0 ∈ GD be any arrow, beginning at some m0 ∈M . By Section 5.3,
m0 lies in some open neighborhood of G contained entirely in GD. In this neighborhood the
connection D is tangent to some foliation F of the neighborhood.
Using the fact that the foliation F is locally fibrating, and using the fact that the restriction
of the target projection β : G → M to each leaf of F is a local diffeomorphism, it follows from
the implicit function theorem that we can find a chart ψ : U × V → G, for a neighborhood
of m0 in G, simultaneously adapted to the foliation and the target projection β. Indeed, we may
take U to be an open subset of m0 in M , V to be an open subset of the target-fibre β−1(m0),
and may arrange that: (i) β(ψ(m, p)) = m; and (ii) ψ(U × {p}) is an integral manifold of D
through p, and the image of a local bisection of G, for every p ∈ V . See Fig. 3.
Figure 3. A chart ψ : U × V → G for a neighborhood of m0 ∈M .
Since g0 ∈ GD, there is a local bisection b of G integrating D with b(m0) = g0 (see Fig. 3).
Shrinking neighborhoods if necessary, we may suppose b has domain U . We now use multi-
plication in the groupoid to translate the chart ψ to a chart ψ̄ : U × V → G whose image is
a neighborhood of g0: Put ψ̄(m, p) = b(m)ψ(m, p), which is well-defined on account of (i) above.
Now ψ̄ maps {m}×V diffeomorphically into a fibre of the target projection. On the other hand,
for each fixed p ∈ V , the map m 7→ ψ̄(m, p) : U → G is an immersion whose image Bp is the
product of the local bisection b(U) and the local bisection ψ(U ×{p}) (here viewing local bisec-
tions as submanifolds of G). The two preceding remarks show that ψ̄ has a tangent map of full
rank, so that ψ̄(U × V ) is an open neighborhood of g0. Furthermore, as the local bisection Bp
is a product of local bisections integrating D, for each p ∈ V , Bp is itself an integral manifold
of D, by Lemma 5.1. Whence ψ̄(U × V ) is an open neighborhood of g0 contained in GD.
5.5 Proof that GD ⊂ G is closed
That GD ⊂ G is closed follows from a general property of involutive n-plane fields: The torsion
tensor τ , a section of ∧2D∗ ⊗ (TM/D) defined by τ(V1, V2) = [V1, V2] mod D, must vanish
on GD. Since τ is continuous, the complement of GD in G is therefore open.
Since GD is simultaneously open and closed in G, it is a union of connected components.
Since the identity bisection integrates any Cartan connection D on G, we have G0 ⊂ G. This
completes the proof of part (2) of Theorem 1.2.
Cartan Connections on Lie Groupoids and their Integrability 25
A The canonical Cartan connection for Riemannian geometry
Here we sketch the construction of the canonical Cartan connection associated with every Rie-
mannian manifold M . Details will be given elsewhere. The detailed construction of the cor-
responding infinitesimal connection appears in [4]. For simplicity, we shall take ‘isometry’ to
mean ‘orientation preserving isometry’. For then the transitive Lie groupoid G ⊂ J1(M ×M)
defined in Section 1.2 will have connected isotropy groups and connected source fibres (M is
assumed to be connected).
Define the prolongation pG ⊂ J1G of G by pG = J2(M ×M) ∩ J1G. Here J2(M ×M)
denotes the Lie groupoid of two-jets of local bisections of M ×M , regarded as a subgroupoid of
J1(J1(M ×M)).
Lemma A.1. The natural projection pG→ G is an isomorphism.
Supposing the lemma to be true, we let S : G → J1G be the unique right-inverse for the
natural projection J1G→ G that has pG as its image. Then S is a Cartan connection defining
an n-plane field D on G.
Proposition A.2. A local transformation φ of M is an isometry if and only if
φ(m) = β(b(m)) (A.1)
for some local bisection b of G integrating D.
Here β : G→M denotes the target projection.
Proof. The local transformations φ of M that are isometries are in one-to-one correspondence
with the holonomic bisections of G (those bisections b that are of the form b = J1φ, for some
bisection φ of M × M). In one direction the correspondence is given by first-order exten-
sion, φ 7→ J1φ. In the other direction we pass from a bisection b to a local transformation φ
using (A.1). But it is a fact that a bisection b of G is holonomic if and only if its extension J1b
is a bisection of pG, i.e., if and only if J1
mb ∈ S(b(m)) at each m in the domain of b. Or, in other
words, viewing b as a submanifold of G, if and only if b is an integral manifold of D. �
Proof of Lemma A.1. The corresponding infinitesimal statement is true. That is, the natural
projection pg→ g, where pg ⊂ J1g is defined by pg := J2(TM)∩ J1g, is an isomorphism. This
follows from the exactness of a natural sequence
0→ ker δ → pg→ g→ coker δ,
where δ : T ∗M ⊗ h→ ∧2T ∗M ⊗TM is Spencer’s coboundary operator and h ⊂ g is the kernel of
the anchor of g (the Lie algebra bundle of skew-symmetric endomorphisms of tangent spaces in
the present case). For details see [4].
To show that pG → G is injective it suffices to show that the pre-image Σ ⊂ pG of the
subgroup of identity elements is trivial, i.e., consists only of identity elements. The Lie algebroid
of Σ is trivial, because it is the kernel of pg → g. That the totally intransitive Lie groupoid Σ
itself is trivial follows from the fact that its fibres are simply-connected Abelian Lie groups, as
is not too hard to show.
Since pg → g is surjective, the morphism pG → G is a submersion. Since its image I is
a wide subgroupoid of G, this image must be open and closed. Since G has source-connected
fibres, G is connected, implying I = G. �
26 A.D. Blaom
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1 Introduction
1.1 Cartan connections on Lie groupoids
1.2 Examples
1.3 Infinitesimal Cartan connections
1.4 The main theorem
1.5 Pseudoactions
1.6 Connections to other work
1.7 Multiplication in J1 G and parallel translation
2 Classical Cartan connections
2.1 Classical Cartan connections
2.2 Cartan connections with model data
2.3 Cartan connections from classical Cartan connections
2.4 Infinitesimalization of a classical Cartan connection
3 The multiplicative structure of J1 G
3.1 Representing elements of J1 G and its Lie algebra
3.2 The adjoint representation
3.3 Recovering a classical Cartan connection
3.4 A linear model of the kernel of J1 G G
3.5 Multiplication and inversion in J1 G
3.6 Source connections and parallel actions
3.7 Multiplication in J1 G in special cases
3.8 Proof of Theorem 3.3
4 On the infinitesimalization of Cartan connections
4.1 Differentiating the parallel action defined by a source connection
4.2 The Lie algebroid of `39`42`"613A``45`47`"603AAut(TM,g)
4.3 The proof of Theorem 4.1
5 Proof of Theorem 1.2
5.1 Proof of Theorem 1.2(1)
5.2 Proof that GD G is a subgroupoid
5.3 Proof that GD contains an open neighborhood of M
5.4 Proof that GD G is open
5.5 Proof that GD G is closed
A The canonical Cartan connection for Riemannian geometry
References
|