Lie algebroids and Cartan's method of equivalence

Elie Cartan's general equivalence problem is recast in the language of Lie algebroids. The resulting formalism, being coordinate and model-free, allows for a full geometric interpretation of Cartan's method of equivalence via reduction and prolongation. We show how to construct certain normal forms (Cartan algebroids) for objects of finite-type, and are able to interpret these directly as infinitesimal symmetries deformed by curvature.' Details are developed for transitive structures but rudiments of the theory include intransitive structures (intransitive symmetry deformations). Detailed illustrations include subriemannian contact structures and conformal geometry.


A new setting for Cartan's me
This paper has its origins in attempts to un guage as possible,Élie Cartan's assertion that fin 'symmetries deformed by curvature.' Having iden structures well-suited to this viewpoint -namel suitably compatible linear connection, called Car problem of realizing this model in practice. To do method of equivalence, and even to reformulate t that it addresses. The advantages of this reform computational, as we shall demonstrate.
The introduction to this paper is in two part Cartan's equivalence problem in the language of cific objectives for the paper. We describe the ba the paper will be concerned: certain infinitesima and Cartan algebroids, which amount to normal In Sect. 2 we outline those elements of Cartan Lie algebroid setting, to associate, with any give sically defined Cartan algebroid. Limitations of remainder of the paper will also be given there.
1.1. The equivalence problem. Cartan's met for determining when two objects are equivalen The method applies to an astonishing variety o polynomials and variational problems; tensor stru as Riemannian, conformal, symplectic, complex a ferential operators and associated smooth manif projective structures; and so on. For an introduc of applications see [15,8,10,2,14].
What makes the method so general is that the in terms of certain secondary data of universal for formation about the objects, rather than in terms tan's original approach the secondary data is a co defined pointwise up to extra 'group parameters.' secondary data is a G-structure (see, e.g., [17]) (see, e.g., [2]). While in practice the construction m ∈ M , vanishing Lie derivative along V . Then is a subbundle g ⊂ J 1 (T M ) and V is a Killing fie only if its first-order prolongation J 1 V is a secti of the form J 1 V for some V are called holonomic (1) The Killing fields of σ are in one-to-one co sections of g. Moreover, as we show later, σ can be recovered f that little is lost by restricting attention to g.
The important observation to make here is th is the tangent bundle T M , and its first jet J 1 (T M language associated with these objects, we have: (2) The bundle g ⊂ J 1 (T M ) of 1-symmetries o isotropy subalgebroid of σ under the represen determined by the adjoint representation of J The terms 'isotropy' and 'adjoint representation' algebroids of familiar Lie algebra notions. The algebroid is described in in Sect. 3

. Isotropy suba
A Lie algebroid over a smooth manifold M is ve a Lie bracket on its space of sections, and a vect called the anchor, satisfying certain conditions m tions of both tangent bundles (# : T M → T M t a single point). The bundle of k-jets of sections o algebroid.
Lie algebroids over M also generalize the infi on M (see 1.7 below), in which case the image o distribution tangent to the foliation of M by orbi arbitrary Lie algebroid is always tangent to some accordingly called orbits; a Lie algebroid with sur 1.3. Infinitesimal geometric structures and geometric structures, as defined below, generalize of a Riemannian metric: Definition. Let t be any Lie algebroid over M simplest case). Then an infinitesimal geometric s metric is surjective. We will see that every Poi infinitesimal geometric structure g ⊂ J 1 (T * M ) not transitive. A simple example of an infinitesim be surjective or transitive is the joint isotropy sub and a vector field V with non-degenerate energy Structures sometimes viewed as transitive are a tion is invariantly formulated. For example, almo cally intransitive structures.
Associated with any G-structure on M is a corr structure on T M , but this structure is always su Here now, in Lie algebroid language, is an equivalence: Equivalence Problem. Given smooth manifol geometric structures g 1 ⊂ J 1 (T M 1 ) and g 2 ⊂ J ists a diffeomorphism φ : M 1 → M 2 , with assoc T M 2 , such that the corresponding Lie algebroid is J 1 (T M 2 ) maps g 1 isomorphically onto g 2 .
Remark. If we want to formulate a more gener T M 1 and T M 2 with general Lie algebroids t 1 and arbitrary Lie algebroid morphisms t 1 → t 2 instead T M 2 , or we must restrict the class of infinitesima so that morphisms of 'coordinate change type' c restricted purposes of the the present paper, furt be unnecessary.

1.4.
Cartan's method. Having formulated the appropriate secondary data, Cartan's method a appropriate normal form. In the original one-f normal form is a coframe, on a possibly larger s eliminated. See, e.g., [8] or [15].
The normalizing algorithm involves two funda duction and prolongation. If the secondary data then reduction amounts to identifying coordinate is understood as a 'symmetry deformed by curv interpretation is often obscured, however.
For objects of infinite-type, the normalizing an altogether different criterion for equivalence m jects can be identified by applying Cartan's 'inv extensively in [2,10]. They will not be studied h 1.5. The symmetries of infinitesimal geom the Killing fields of a Riemannian structure are th geometric structure g ⊂ J 1 t. These are those sec J 1 V ⊂ J 1 t are sections of g. Evidently, the sy correspondence with the holonomic sections of g prolongations of something. Symmetries are nece and are closed under the Lie algebroid bracket.
We will not be presenting a complete solution Rather, our main focus is the following: Obstruction Problem. Given an infinitesimal the obstructions to the existence of symmetries o We now turn our attention to the normal for out of infinitesimal geometric structures. We be form explicitly in the case of Riemannian geometr curvature invariants enter as the obstruction to s 1.6. A normal form for Riemannian geom bundle of 1-symmetries of a Riemannian metric σ that the Levi-Cevita connection ∇ associated with of a canonical exact sequence, We have a corresponding exact sequence, where h ⊂ T * M ⊗ T M denotes the o(n)-bundle space endomorphisms, which is ∇-invariant. By t subbundle of J 1 (T M ) lies inside g and we obtain (1) With the help of the Bianchi identities for line curv ∇ (1) implying that ∇ (1) is flat if and only if curv ∇ is Now h-invariance implies, by purely algebraic ar scalar component; ∇-invariance then implies co from (1) one recovers the standard criterion for Riemannian manifold.
1.7. Cartan algebroids: symmetries deform nection ∇ on a Lie algebroid g is a Cartan conn with the Lie algebroid structure [1]. The pair ( The formal definition and basic properties are re In the Riemannian example above, the pair (g we saw that g ∼ = g 0 × M , for some Lie algebra fact, whenever g 0 is any Lie algebra, acting smo trivial bundle g 0 × M inherits the structure of a transformation) algebroid, and the trivial flat co Conversely, any Cartan algebroid with a flat Cart algebroid (Theorem 4.6, Sect. 4). It is in this infinitesimal symmetries deformed by curvature. M algebroid may be regarded as deformations of orb In [1] we described how Cartan algebroids may free, and possibly intransitive versions of classic tioned other alternative models contained in the has delineated the relationship between transitiv tractor bundles [4], which like Cartan algebroids like them are based on a transitive model fixed a One consequence of choosing a model-free ap Generally, 'curvature' has referred to the local de -typically R n or a homogeneous space G/Happroach, such as the one described here, all po and curvature merely measures the local deviation space. From this point of view, Euclidean space, h 2.1. Cartan connections via generators. To mulated in Sect. 1, we attempt to reduce it to a s below. To formulate the result, recall that the bundle t are in one-to-one correspondence with th exact sequence 0 → T * M ⊗ t ֒→ J 1 t → Now suppose t is a Lie algebroid and g ⊂ J 1 t an Then we call ∇ a generator of g ⊂ J 1 t if s(t 1 ) image of g. Generators are certain 'preferred co but need not be unique. For example, the Levifor the bundle g ⊂ J 1 (T M ) of 1-symmetries of a any linear connection ∇ on M such that∇σ = 0 Generators are indispensable in explicit computa The following crucial observation is not diffic Sect. 6.) Theorem. Let g ⊂ J 1 t be an infinitesimal geo projection a : g → t has constant rank. Then g only if it is surjective and has structure kernel h connection on t whose parallel sections are precis In particular, curv ∇ is then the local obstruction When geometric structures do not satisfy the tries to correct this with an appropriate sequen operations described next.

2.2.
Prolongation. The prolongation of an infin J 1 t is a natural 'lift' of g to a subset g (1) ⊂ J 1 g: J 1 (J 1 t), and the existence of a natural inclusion It turns out that g (1) is an infinitesimal geometric constant rank. Most importantly, there is a on symmetries of g and symmetries of g (1) , furnished Proposition. A section W ⊂ t is a symmetry symmetry of g (1) .

2.3.
Reduction. Let g ⊂ J 1 t be an infinitesimal tion of g we shall mean any subalgebroid g ′ ⊂ g it suffices to check that symmetries of g are sym longation, there is no unique way to construct re g ′ ⊂ g is a reduction and g ′′ ⊂ g merely a subalge g ′′ is automatically a reduction of g also. We say We now describe the most important reduction and Θ-reduction.

Elementary reduction.
Returning to Car emphasize that transitivity is not a hypothesis o algebroids can be intransitive). Rather, one requi geometric structure g ⊂ J 1 t is not surjective, w passing to the elementary reduction g 1 of g. By d where t 1 ⊂ t denotes the image of g. Assuming rank, they are subalgebroids. In particular, g 1 geometric structure. Moreover, one easily proves Proposition. If the elementary reduction g 1 of reduction of g in the sense above. If g is surjec g 1 = g then g is contained in J 1 t 1 and is surject structure on t 1 .
Because surjectivity is built into the definitio duction never appears in that setting. Elementar Sect. 7, together with a cruder alternative called 2.5. Θ-reduction. If an infinitesimal geometric s tive but has a non-trivial structure kernel, then, 2.1, one can try to shrink the structure kernel by the prolongation g (1) generally fails to be surjectiv to correct this by turning to the elementary reduc that is computationally more attractive. One a of g (1) by first replacing g by its Θ-reduction. B g (1) ⊂ J 1 g, i.e., the set g (1) 1 := p(g (1) ) ⊂ g, where 2.6. A specific normalizing algorithm and a specific algorithm for constructing a Cartan geometric structure of finite-type. First, we defin By Proposition 2.4, the following procedure, wh g to be surjective: do while g is not surjective replace g with g 1 (elementary reduction) end do.
Next, we let strongly surjectify g denote the follow simultaneously surjective (by Propositions 2.4 an do while g (1) is not surjective surjectify g replace g with g 1 (Θ-reduction) surjectify g end do.
To describe an implementation of this procedur Θ-reduction in the special surjective case.
One might attempt to normalize an infinitesi ementary reduction and prolongation alone. In easier to apply the following algorithm: surjectify g repeat until stop encountered if h = 0 apply Theorem 2.1 and stop strongly surjectify g if h = 0 apply Theorem 2.1 and stop replace g with g (1) (prolongation) end repeat. Notice that prolongation is delayed as long as po in which the above algorithm can fail.
Firstly, an execution of surjectify g or strongly s some iteration of these procedures' do-while loop While prolongation of g might resolve this kind constancy), this requires a prolongation theory algebroid. Also, one needs to understand how co structure combine with transverse information to tunately, a splitting theory for Lie algebroids exist transverse problem to the case of an isolated sin None of this is explored here either.
If the Cartan algorithm above succeeds it ends this delivering a Cartan algebroid whose parallel correspondence with the symmetries of g. We the Cartan algebroid.
2.7. Paper outline. In Sect. 3 we review basic lish attendant notation. In particular, we desc (Koszul) connections afforded by Lie algebroids, deformations of Lie algebroid representations, wh definition of the adjoint representation of J 1 g on the bracket on J 1 g explicitly. We introduce the which are ubiquitous throughout, and developed Sect. 4 summarizes features of Cartan algebro particular, the result that Cartan algebroids are d metries (Theorem 4.6).
Sect. 5 gives many examples of infinitesimal ge isotropy subalgebroids associated with various st We also explain how to associate an infinitesimal g algebroid or a classical G-structure. From our it will be clear how one may associate an infini an arbitrary (but suitably regular) differential o in practice how one computes the image of an without resorting to local coordinate calculations Associated with an infinitesimal geometric st kernel h and image t 1 , is an exact sequence A generator ∇ of g, as defined in 2.1 above, abo sequence, determining an identification g ∼ = t 1 ⊕ tations. Sect. 6 characterizes the linear connectio a more linear presentation may skip to Sect. 11 im 9.5 and the remainder of the paper thereafter.
We have not attempted substantially novel ap the present work. In particular, our application fairly superficial. We hope to correct this deficie of making comparisons with other approaches, subriemannian contact three-manifolds in Sect. 1 is to be found in [9,14]. In addition to construct go on to construct the invariant differential opera In Sect. 11 we return to prolongation, explain generator, and hence how compute prolongations the general case t = T M , but we must assume t is A detailed section on conformal geometry, Sect prolongation results.

Preliminary not
For an introduction to Lie groupoids and algeb tions in this paper are made in the C ∞ category.
Notation. We use Alt k (V ) ∼ = Λ k (V * ) and Sym k ( of R-valued alternating and symmetric k-linear m notation applies to the tensor algebra of a vector of E, then this is indicated by writing σ ∈ Γ(E) o means σ is an E-valued differential two-form on 3.1. Lie algebroids. A Lie algebroid over M c M , a Lie bracket [ · , · ] on the space of sections Γ( morphism # : g → T M , called the anchor. One r Leibnitz identity, where f is an arbitrary smooth function. The 3.2. The definition of gl(E) for a vector b Lie algebra g is a vector space E, together wi g → gl(E) := Hom(E, E). Turning now to the g Lie algebroid representations, let E be a vector b consider the exact sequence  Remark. gl(E) is in fact a realization of the L GL(E) of isomorphisms between fibres of the vect on E. So elements of gl(E) have the interpretatio For details, see [12] (where gl(E) is denoted D(E Suppose X is a section of g. When the section ∇( be viewed as a differential operator, we instead wri In view of the preceding characterization of the se have the Leibnitz identity

Conversely:
Proposition. Every vector bundle morphism ∇ nitz in the above sense is a g-connection.
If ∇ is a g-connection, then the formula The g-connection ∇ is a g-representation when c Example. If g is a Lie algebroid and E ⊂ g is kernel of its anchor then a canonical representa by ρ X Y := [X, Y ] g . Important cases in point a and the structure kernel of an infinitesimal geom constant rank.
3.4. Linear connections. Using the language of connection ∇ on E is just a T M -connection on E when ∇ is flat. It is an elementary fact that th one-to-one correspondence with the splittings s : The splitting associated with a linear connection given by The adjoint representation. The general resentations to a Lie algebroid g is not a self-repr tation of J 1 g on g. This representation is well-de shows that ad g is indeed a representation (an We note that a ad g ξ X = ad h (J 1 a)ξ (aX); 3.7. The bracket on J 1 ( · ) of a Lie algebr J 1 g is implicitly defined by the requirement 3.6( representation, we now describe this bracket conc Although the exact sequence possesses no canonical splitting, the correspondin is split by J 1 : Γ(g) → Γ(J 1 g), delivering a canon Under this identification, the Lie algebra Γ(J 1 g) in the proposition below.
In addition to having the adjoint representati sentation of J 1 g on T M , given by the composite So we can construct a natural representation of J To prove the proposition one uses 3.6(1) and the finitely generated by those of the form df ⊗ X = 3.8. Dual connections, torsion, and associat algebroid and ∇ a g-connection on itself. We d connection ∇ * on g defined by The torsion of ∇ is the section, tor ∇, of Alt between ∇ and its dual: The torsion or curvature of ∇ can be expressed in of ∇ * (and, by duality, vice versa): Proposition. Let ∇ be a g-connection on g, and We now introduce two important connections connection on T M . They are examples of asso generally in 6.3.
Let ∇ be an arbitrary linear (i.e., T M -) conn associated g-connection on g is defined bȳ The associated g-connection on T M is defined bȳ 4.1. Action algebroids. Let g 0 be a finite-dim [ · , · ] g0 acting smoothly on a manifold M . Tha homomorphism ρ : g 0 → Γ(T M ). We may rega itesimal symmetries. The trivial bundle g := g 0 algebroid structure. This is the associated action The anchor of this Lie algebroid is the 'action (ξ, m) to ρξ(m). The Lie bracket on Γ(g 0 × M ) is on g 0 , regarded as the subspace g 0 ⊂ Γ(g 0 × M ) it, let τ ( · , · ) denote the naive extension of this br with respect to all smooth functions (and conseq And let ∇ denote the canonical flat connection o on Γ(g 0 × M ) is defined by Notice that if∇ denotes the associated g-connec 4.2. Cartan connections. Let ∇ be a linear Then ∇ is a Cartan connection if the correspond of the exact sequence of Lie algebroids, is a Lie algebroid morphism. A Cartan algebroid a Cartan connection. A morphism of Cartan a preserving morphism of the underlying Lie algeb It follows immediately from the definition th T M , associated with a Cartan connection ∇, a In particular, every Cartan algebroid g has a ca converse statement, see the corollary below.
where∇ denotes the associated g-connection T M in the second. As simple consequences of (4) we have: Corollary. (6) Suppose g is transitive. Then ∇ is a Cartan associated g-connection∇ on g is flat. The symmetric part of a Cartan algeb broid g has a canonical subalgebroid isomorphic t let ∇ denote the Cartan connection and let g 0 parallel sections, which is finite-dimensional. Th that g 0 ⊂ Γ(g) is a Lie subalgebra, and we obtain Equipping the action algebroid g 0 × M with its ca a morphism of Cartan algebroids, Assuming M is connected, this morphism is inje vanishing at a point vanish everywhere. We call (1) the symmetric part of g.

4.6.
Curvature as the local obstruction to g is globally flat if it is isomorphic to an action its canonical flat connection -or, equivalently, part. We call g flat if every point of M has an op restriction g| U is globally flat. 3 The following theorem shows that a Cartan infinitesimal symmetry deformed by curvature.
Theorem ([1]). Let g be a Cartan algebroid wi over a connected manifold M . Then g is flat if a is simply-connected, flatness already implies globa In the globally flat case the bracket on the Lie is given by where∇ denotes the associated representation of The necessity of vanishing curvature is sertions in the first paragraph it suffices to sho For the application of Theorem 4.6 to examples 4

Examples of infinitesimal geo
In this section we describe the infinitesimal with Riemannian structures, vector fields on a R almost complex structures, Poisson structures, group, affine structures, projective structures, cl algebroids.
Subriemannian contact structures and conform rately in Sections 10 and 12. Conformal parallelis 5.1. Isotropy. Most infinitesimal geometric stru understood as isotropy (or joint isotropy) subalgeb sentations. In the case of Riemannian geometry it metric σ ⊂ Sym 2 (T M ), i.e., of a section of som ometry, it is the isotropy of a rank-one subbundl geometry, it is the isotropy of an affine subbundl following definition of isotropy is general enough Let ρ : g → gl(E) denote some representation denote any affine subbundle of E (a single sectio Σ 0 ⊂ E the corresponding vector subbundle para Assume either that Σ 0 is g-invariant or that Σ = collection of all elements x ∈ g for which The isotropy of Σ is a subset of g intersecting sions may vary, i.e., is a 'variable-rank subbundle' under the bracket of g. When this rank is consta bundle and consequently a subalgebroid, called t The structure kernel of g is the isotropy h ⊂ T * M representation. So h is the bundle of σ-skew-sym spaces, a Lie algebra bundle modeled on o(n), conformally equivalent metrics give the same stru One way to see that g is surjective (i.e., transitiv B.1 in Appendix B to the morphism X → X · σ kernel is g. On account of the surjectivity of the re Sym 2 (T M ) of this morphism, the lemma delivers is surjective and has constan and thus an infinitesimal geometric structure). The lemma just applied is very useful in dete kernel of infinitesimal geometric structures defin plications of Lemma B.1 are made in 5.3 and 5 accompany subsequent applications.
The symmetries of g (in the sense of 1.3) are t vanishing Lie derivative, i.e., its Killing fields. A connection ∇ on T M such that σ is∇-parallel, The Levi-Cevita connection is thus the unique to From g one can recover the metric σ up to a conformal class). In the simply-connected case, s Proposition. Let h ⊂ T * M ⊗ T M denote the o bitrary conformal structure. Then on simply-con surjective infinitesimal geometric structure g ⊂ J is the isotropy subalgebroid of some Riemannian s class. This structure is uniquely determined up to Proof. Suppose g ⊂ J 1 (T M ) has structure kern the line bundle determined by the conformal str of h-invariant elements of Sym 2 (T M ). The non-v positive or negative definite. By Lemma B.2, L section σ, unique up to constant. Changing the the sought after metric.
The application of Cartan's method to Riema by g ⊂ J 1 (T M ) as above, g acts on T M by restr isotropy is the isotropy g V ⊂ g of V .
The structure kernel of g V is the o(n − 1)-bu space endomorphisms infinitesimally fixing V (m In particular, g has constant rank (is an infinite only if V has constant length or 1 2 V 2 is a free (transitive) in the former case only.
Proof. Applying Lemma B.1 to the morphism X the kernel, we deduce that D is the kernel of the m diagram commute: Let ∇ be any generator of g (e.g., the Levi-Ce g the corresponding splitting of (2) above. Th Proof. First, note that i.e., θ(X)U = ad X (JU ) − Applying Lemma B.1 to this morphism, we obtain kernel, and satisfying Although not of finit More generally, (1) defines a Lie algebroid structu fold (M, Π), with anchor # defined by (2). The sy the orbits of the Lie algebroid T * M . An infinitesimal isometry of a Poisson manifo M such that L V Π = 0. Poisson manifolds ha isometries. In particular, every closed 1-form α isometry #α tangent to the symplectic leaves kno field, or a Hamiltonian vector field if α is exact.
It is not too difficult to establish the following

5.7.
Subgeometries of an Abelian Lie group of a geometric structure which has, in general, triv Lie group, V its Lie algebra, and M ⊂ E a cod be the V -valued one-form on M obtained by rest E. Then dω = 0 and dim V = dim M + 1. Let Any suitably non-degen tial operator on M , defines an infinitesimal geom J 1 (J k (T M )). As a simple example, which will su ciple, we consider an affine structure on M , i.e., on M , in which case k = 1. The relevant nonisotropy of the torsion of ∇ should have constant View an affine structure ∇ as a section of J 1 ( In order to associate a natural isotropy subalgebr servations. First, J 1 (J 1 (T M )) acts on J 1 (T M ) * ⊗ acts on J 1 (T M ) via adjoint action, and on T M v a condition that is second-order in U . Unravellin tions defined above, we may write this condition It easily follows that J 1 U is a symmetry of g when of ∇. Suppose, conversely, that X ⊂ J 1 (T M ) is a s in g. This means: Here b is the restriction of the canonical projec establishes (2). 5.9. Projective structures. Recall that two lin tively equivalent if their geodesics coincide as unpa their difference ∇ − ∇ ′ , which may be viewed as [11]. In particular, P is a princip is a transitive Lie algebroid over M , and the as representations of g; see, e.g., [12]. As P is a f representation (see below). That is, we have a Li This turns out to be injective, identifying g with infinitesimal geometric structure on T M is surjec The representation of g on T M may be describ of g := T P/G with G-invariant vector fields on P on P to identify sections V of T M with G-inva Then X · V := L X V , where L denotes Lie deriva 5.11. Cartan algebroids as infinitesimal g seen that all surjective infinitesimal geometric str nel define Cartan algebroids (Theorem 2.1). Con with Cartan connection ∇, then g := s ∇ (t) ⊂ J 1 metric structure generated by ∇ with trivial stru the splitting of 0 → T * M ⊗ t ֒→ J 1 t → determined by ∇.

Generators, associated operators
Picking a generator for an infinitesimal geomet identify g with the direct sum t 1 ⊕ h of its image greatly facilitates computations. Generators are contact three-manifolds.) In principle, any invar expressed in terms of associated differential opera tive infinitesimal geometric structures. In 6.4 w derivative, and in 6.5 analogues of the classical B 6.1. Basic properties of generators. Let g ⊂ structure, with structure kernel h, and image t 1 constant rank if and only if h ⊂ T * M ⊗ t (or equi (i.e., are subalgebroids).
Proposition. If g a − → t has constant rank then: (1) g admits a generator ∇.
We recall that cocurvature was defined in 4.3.
6.3. Associated connections and differentia infinitesimal geometric structure with structure generator of g. Then for each representation E of phism ρ : g → gl(E), we have an associated t 1 -co on E if g is surjective). By definition, this is t where s ∇ : t → J 1 t is the splitting of 6.1(4) corre

Examples.
(1) Taking g := J 1 t and ρ = ad t , we obtain This is the associated t-connection on t defin (2) Let g := J 1 t act on T M via the composite Then we similarly computē This is the associated t-connection on T M d (3) An arbitrary infinitesimal geometric structu The associated derivative of a g-tensor σ ∈ Γ(E where∇ is the associated t 1 -connection on E. A we mean that t 1 ⊂ t is invariant under the adjoin for example, if g is surjective. (Image reduction g-representation, implying∇σ is another g-tenso closed under associated derivative. In particular, obtain higher order differential operators.
Additionally supposing that all g-representatio into g-representations coming from some collecti we have finit for some n ij ∈ I, and obtain a corresponding dec We call the differential operators ∂ ij : Γ(E i ) → Γ( ferential operators; all differential operators which out of associated connections∇ are combination If there is a canonical way in which to choose th differential operators become invariant different infinitesimal geometric structure g. Significant ca (5) The case where t is a Cartan algebroid discuss are just t-representations because g ∼ = t. (6) The case where the generator ∇ of g is uniq ducing the situation to case (5) above. (7) The case where torsion tor∇ has a natural 'n For invariant differential operators associated wi manifolds, see Sect. 10.
6.4. The associated exterior derivative. Let metric structure with structure kernel h. Then degree k is a section θ ⊂ Alt k (t 1 ) ⊗ E, where t 1 g-representation. (We use t 1 , rather than t, to derivative d∇θ ⊂ Alt k (t 1 ) ⊗ E of θ is defined in t (2) For any g-type differential form θ, we have Here the wedge implies a contraction φ ⊗ σ → representation of h on E.
Proof of (2). The general case can easily be red prove now. Letting s : t → J 1 t denote the splitti compute, for arbitrary U 1 , U 2 ⊂ t 1 , 6.5. Bianchi identities. Generalizing the classic below exhibit certain algebraic and differential rooted in the equality of mixed partial derivativ Next, assume g admits a representation E for w h → gl(E) is faithful (injective), and let θ ⊂ E differential form of degree zero. Then, combining proposition, we obtain d 3 ∇ θ = d∇Ω ∧ θ + Ω ∧ d ∇ conclude that d∇Ω ∧ θ = 0. Since θ is arbitrar obtain A little manipulation allows us to write (1) and (

Elementary reduction and
In this section we study elementary reduction, call image reduction. These techniques are usef ric structures fails to be surjective, and in parti geometric structures on T M . A simple applicati mannian three-manifold is included. 7.1. Image reduction. Let g ⊂ J 1 t be an infin structure kernel h ⊂ T * M ⊗ t and image t 1 ⊂ t. constant rank. Then the image reduction of g is t 1 ⊂ t, under the adjoint representation of g ⊂ that image reduction is cruder than elementary described further below. Nevertheless, it is usuall and this may simplify the subsequent application Define J ⊂ T * M ⊗ T M by JU := n × U , where restricts to a complex structure on level sets of f a rank-one structure kernel spanned by J. Using that its image is T = ker df ∩ ker dE. We ther i.e., the joint isotropy, g := (J 1 (T M )) σ,f,df, T ⊂ We observe that g has trivial structure kernel, h this one applies Lemma B.1 to the morphism, which has g as kernel.
As g itself is evidently stable under image-redu reduction. By Proposition 6.1, g has a generator g = (J 1 (T M )) σ,f,df, T , we must havē

Prolongation and
In this section we characterize the prolongati isotropy of a tautological one-form a and its 'to logues of classical objects bearing the same nam valid when g is transitive. We begin, however, wi subbundle J 2 g ⊂ J 1 (J 1 g) that is completely gene This section concludes with the reformulation associated with torsion. 8.1. Prolongation. Let t be an arbitrary vecto a natural inclusion of vector bundles J 2 t ֒→ J J 1 (J 1 W )(m); W ⊂ J 1 t. As a basic fact one has Lemma. For any section X ⊂ J 1 t, X is holonom Now the definition of prolongation, g (1) := J 1 g ∩ makes sense in general, but suppose for the mom g ⊂ J 1 t is an infinitesimal geometric structure subalgebroid, implying g (1) is an infinitesimal ge g (1) has constant rank. Let W ⊂ t be a symme consequence of definitions is that J 1 W is a sectio of g (1) . In fact, it is a consequence of the lemma arise in this way: for arbitrary smooth functions f on M . The above construction, holding for arbitrary t case that t is replaced by J 1 t. This delivers an o which will also be denoted D. In the formulas ab by the natural projection p : Proposition (Characterization of J 2 t ⊂ J 1 (J 1 vector bundle t, one has J 2 t = ker ω 2 , where In this proposition some D's are operators Γ(J 1 t operators Γ(J 1 (J 1 t)) → Γ(T * M ⊗ J 1 t). All ambi Since the proposition above is just a general fa is relegated to Appendix B.4.

Torsion.
We now return to the case that g i ture on a Lie algebroid t. Applying the general terization of g (1) as an isotropy subalgebroid.
Regard the restriction a : g → t of J 1 t → t as this is the tautological one-form. The adjoint rep a representation of g on t. So the exterior deriva g-form, of degree two. This is the torsion of the s Remark. If g is intransitive, then (J 1 g) a,da gen be the prolongation of g: every section of h ⊂ T * M in the image of the anchor # : g → T M turns o that is not a symmetry of g (1) .
Proof of lemma. Begin by observing that (ξ · a)(X) = ad t pξ (aX) − Since a : g → t is a Lie algebroid morphism, the id ad t (J 1 a)ξ (aX), and so (ξ · a)(X) = ad t pξ−(J 1 a) Note here that J 1 a : J 1 g → J 1 t is the morphism Because pξ −(J 1 a)ξ is a section of the kernel of J 1 of T * M ⊗ t and, applying 3.6(2), obtain It is not too difficult to show that ξ · da = d Therefore . Normalizing torsion and the upper co g with t ⊕ h by choosing a generator ∇ of g. identification, and a corresponding splitting of the torsion da = tor∇ ⊕ ev ⊕ Here∇ denotes the associated t-connection on t φ) := φ(V ). Notice that tor∇ is the only compon of generator. Given two generators ∇ 1 and ∇ 2 , viewed as a section of t * ⊗ h and one readily com where ∆ denotes the upper coboundary morphism Here # * : T * M → t * is the dual of the anchor # : As an elementary consequence of (1) above, we o Proposition. If C ⊂ Alt 2 (t) ⊗ t is a compleme exists a generator ∇ such that tor∇ ⊂ C. If ∆ unique.
Note that there is no need to require that C be g Assumption. In this section g ⊂ J 1 t is a surjec ture over a transitive Lie algebroid t. In particula tive, g has a structure kernel h of constant rank a tion 6.1(1)).
Our chief objective is a characterization of the Θ-r an explicit knowledge of g (1) .
9.1. The lower coboundary morphism. As in morphism' plays a central role in Θ-reduction. H coboundary morphism ∆, defined in 8.4, is not need the lower coboundary morphism δ, defined This morphism is also As we assume t is transitive, we may, by dualiz regard T * M as a subbundle of t * , and obtain nat With this understanding, we may regard δ : T * restriction of the upper coboundary morphism ∆ 8.4.
The analogue of the torsion bundle H(g) defi bundle h(g) := Alt 2 (T M ) im δ Whenever h(g) is a genuine vector bundle (has con There is evidently a natural morphism ψ : h(g) → gram commute: (1) where p : J 1 g → g is the projection. This follow e.g., 8.3(3). One establishes (1) by applying Lem Now g (1) is the kernel of the morphism ξ → ξ · da it follows from 8.3 (2) and transitivity that: (2) For any ξ ∈ (J 1 g) a , the element ξ · da ∈ Alt 2 ( an element (ξ · da) ∨ ∈ Alt 2 (T M ) ⊗ t. This means we may regard g (1) as the kernel of a According to (1), the domain of θ fits into an exa Applying Lemma B.1 to the morphism θ, we obtai 0 → ker δ ֒→ g (1) a (1) −−→ ke where Θ is the unique morphism making the foll Proposition. If g ⊂ J 1 t is surjective and t is t morphism Θ : g → h(g), constructed above, such 0 → ker δ ֒→ g (1) a (1) −−→ g is exact. In particular, the structure kernel of g (1) ary morphism δ, while the image g (1) 1 of g (1) (th of Θ. If ker δ and ker Θ have constant rank the geometric structure.
Remark. By the proposition the structure kerne h ⊂ T * M ⊗ g and is consequently commutativ Corollary. Suppose that the torsion bundle H( torsion reduction g τ of g is well-defined. Then g (1 1 coboundary morphism δ has constant rank, and bo then g τ is a reduction of g in the sense of 2.3. torsion reduction coincide.
Here the rank hypotheses and Proposition 9.2 ens that Proposition 2.5 applies. However, the result rank hypothesis on g τ alone.

Structures both surjective and Θ-redu reduced if it coincides with its Θ-reduction.
Theorem. Let g ⊂ J 1 t be a surjective infinitesim sitive Lie algebroid t. Assume that g is Θ-reduc defined above vanishes). Assume that the associa is injective. Then g has an associated Cartan al with a canonical Cartan connection ∇ (1) . The ∇ with the prolonged symmetries of g.
Proof. Proposition 2.5 implies the prolongation g g (1) has trivial structure kernel, because we suppo Applying Theorem 2.1 to the infinitesimal geom Cartan connection ∇ (1) on g whose parallel sec These are nothing but the prolonged symmetries In Proposition 11.1 we characterize ∇ (1) as th g whose curvature curv ∇ (1) ⊂ Alt 2 (T M ) ⊗ g * ⊗ formula expressing ∇ (1) in terms of a generator o 9.5. The special case t = T M . When t = T M , are the same thing, as are the upper and lower c We now rewrite the above theorem accordingly, a the Cartan connection that we establish later in Here∇ will denote the dual of ∇, i.e.,∇ U V J 1 (T M ) reductive if ∆ has constant rank and if th complement C. We call the generator ∇ norma Proposition 8.4 guarantees the existence of norm (2) Identifying g with T M ⊕ h using the generat for any normal generator ∇.
When one of the conditions in (3) holds, obst ularly simple to describe, as is the symmetry L case. Indeed, one then computes, with the help o d∇ curv∇ = 0, Here ∇ (1) denotes the representation of g on i connection ∇ (1) on g. Applying Theorem 4.6: Corollary. Let g ⊂ J 1 (T M ) be an infinitesimal hypotheses of the above theorem, and assume ei normal, or that τ = 0 and ∇ is torsion-free. L set and g 0 be the Lie algebra of all symmetries o U is simply-connected, then equality holds if and and∇-parallel. In that case g 0 is naturally isom arbitrary) with Lie bracket given by 9.6. The symmetries of Riemannian structu bundle of 1-symmetries of a Riemannian metric The upper coboundary morphism for g is a map According to Corollary 9.5, we are in the maxim is both h-invariant and ∇-parallel. According theoretic analysis of the curvature module, this h for some constant s ∈ R (the scalar curvature). described in the corollary is then isomorphic to isometries of Euclidean space, hyperbolic space, o s = 0, s < 0, or s > 0. The upper boundary morphism, given by . We leave it t vanishing intrinsic torsion τ precisely when for some one-form α. While α depends on the two-form dα does not.
Assuming τ = 0, g has a unique torsion-free definition of τ ). Moreover, it is not hard to show ∇ U ω = α(U )ω, U and accordingly that curv∇ = dα ⊗ id Applying Corollary 9.5, we are in the maxima Here we shall understand H to be transversal specification of the subriemannian structure to in orientation. This amounts to the choice of a non θ annihilating H. The contact hypothesis means to a symplectic structure on H.
The infinitesimal isometries of the subriemann metries of the infinitesimal geometric structure The subriemannian metric σ has a canonical ex metric, defined as follows: Let n be the Reeb normalized contact form θ. That is, dθ(n, · ) = 0, θ(n One extends σ so as to make n orthogonal to H σ(n) := σ(n, · ). The easy proof of the following Proposition. The reduction g ⊂ J 1 (T M ) above of the extended metric σ and n, g = J 1 (T M ) σ,n .
10.2. The complex structure on H. Let × den mined by the extended metric σ and define J ⊂ T J has kernel n , image H, and the restriction of on H relating the area form dA to the subrieman The proposition is established by analyzing detail, identifying a natural g-invariant complem Proposition 8.4. This analysis is not hard but to Appendix B.3. For the interested reader, we normalized generator ∇ in terms of the Levi-Cevi extended metric σ.
10.4. Bianchi Identities and low weight di write down Bianchi identities for the normalized invariant differential operators, the systematic co We are now ready to define two invariant op ∂ − : Γ(H 2 ) → Γ(H) according to Associated with the normalized generator ∇ of ants T := tor∇ and Ω := cocurv ∇ = − curv∇, w 6.5(3) and 6.5 (4). Of course these are also invaria structure. According to 10.3(4), T depends only o ant ∇n. As it turns out, one component of Ω is a that Ω(U 1 , U 2 ) ⊂ h for all U 1 , U 2 ⊂ T M (Propo Alt 2 (H), there is a real-valued function κ well de Proposition (Bianchi identities).
(3) ∂ n κ = − 1 2 curl H (∂ − (∇n)). (4) The cocurvature of ∇ is given by Proof. Proposition 10.3(4) states that T (U 1 + a 1 n, U 2 + a 2 n) = (a 1 ∇ U2 n − a A little multilinear algebra determines that Ω ha Ω(U 1 + a 1 n, U 2 + a 2 n)(U 3 + a 3 n) = − κ dA(U for some section ω ⊂ H * and some κ as above. 6.5(4) are equations in bundle-valued three-forms vanishes if and only if λ(U 1 , U 2 , n) = 0 for all se fact to the Bianchi identities gives Suppose that ∇n = 0. Then n is automaticall infinitesimal isometry of the subriemannian contac of 6.1 (3). Note that if the rank-one foliation ge surface Σ, then the invariant function κ drops t In any case, Theorem 9.5 applies, because of 10 − curv∇ above, one applies this theorem and its Proposition (Compare with [9]). Suppose ∇n associated Cartan algebroid, namely g itself. If U and g 0 the Lie algebra of all infinitesimal isomet structure on U, then dim g 0 ≤ rank g = 4. If U holds if and only if the function κ defined by (1 g 0 ∼ = b × R (direct product) where b is the Lie a (Killing fields) of the Euclidean plane, hyperbol whether κ = 0, κ < 0, or κ > 0. 10.6. Invariant differential operators. The n associated invariant differential operators, as expl be invariants of the subriemannian contact struc gradient, curl, etc., of a Riemannian three-manifo Noting that the structure kernel h ⊂ g of g is ducible representations of g by mimicking a know representations of the Lie algebra o(2). At least l for all irreducible representations of g.
Define H 0 := C × M , H 1 := H, and define H 2 we define H k := Sym k−1 C (H) ⊗ C H whereH is H with the complex structure −J. representation and a complex line-bundle, the tw ing to ad J q = kiq; q ⊂ H k Every H k is irreducible as a (real) g-representatio of the irreducible trivial representation R × M .
Recall that for each section q ⊂ E of an irr objective is to derive the decomposition of∇q ⊂ for k = 1. In the latter case we are using the He A compatible pair of splitting morphisms H k+1 as follows: for k = 1 (q a C-valued function). Let q be a section of H k . Then we have a rest splits, we have H * ⊗ H k ∼ = H k−1 ⊕ H k+1 and ar and ∂ − q := π − (∇q|H). That is, ∂ + q ⊂ H k+1 and (∂ + q)(U, V 1 , . . . , V k−1 ) = 1 2 (∇ U q)(V 1 , . . . , V k− for any k 1, for k 2, and ∇ U1 q, U 2 − ∇ U2 q, U 1 = dA for k = 1. This last formula simply means, for q Finally, for any section q ⊂ H k and any k 1 section of H k .
Combining our observation H * ⊗ H k ∼ = H k−1 T M = H ⊗ n , we now obtain: Proposition. For any k 1, we have a natural We assume throughout that t is a transitive Lie see Sect. 9 under 'Assumption.' We continue to d h, and the associated lower coboundary morphism 11.1. Natural connections. Call a linear conne associated g-connection on t -see Example 6.3(4 of g ⊂ J 1 t on t; in symbols, if Here a : g → t is the restricted projection J 1 t → not immediately useful in computations but is stands between the rather abstract Proposition 9 Theorem 11.2 given later.
Corollary. If g ⊂ J 1 t is a Θ-reduced infinites linear connection D on t generates g if and on Corollary. The Cartan connection ∇ (1) in Theo nection on g such that curv If ∇ is a connection on t generating g and ∇ then, identifying g with t ⊕ h using the gener on g is of the form Proof. Let D be the deviation operator described Here s is the splitting in (5). Combining this wit aD #V X + [aX, V ] t − ad t X V = aD #V The claim in (4) now follows from 8.3(1) and tr derived as a consequence of 8.3 (2) and transitivity in (6) is natural and, with the help of (4) and covered, establishing (6).
Proof. Let D denote the general form of a natura with ǫ : t ⊕ h → T * M ⊗ h completely arbitrary. Proposition 11.1, then one computeṡ whereΘ is the morphism defined by (1). Conclus from Proposition 11.1(1). Conclusion (3) is just a (4) by taking ∇ (1) := D; choosing ǫ as described g in Proposition 11.1(3). If Θ = 0 then every gene Note that cocurv ∇(aX, · ) is a section of T * M ⊗ Proof. Since t is assumed to be transitive, there e such that ∇ h #U =∇ U for all U ∈ t. Here∇ deno h discussed in 6.3 (3). After a little manipulation  (1) and (2) into the definition 11.2( Under the identification g ∼ = t ⊕ h determined by stated formula. Proof of Theorem 9.3. By Theorem 11.2(2) and have, ψ(Θ(X)) = i(Θ(X)) mo where i : Alt 2 (T M ) ⊗ t → Alt 2 (t) ⊗ t denotes the being injective). The proposition above then give ψ(Θ(X)) = X · tor∇ mod i 11.4. The special case g ⊂ J 1 (T M ). We now case t = T M . As an application, we complete th unproven assertion of preceding sections.
Proof. In 11.2 above take t = T M and let ∇ h be th with the generator ∇ (given by 6.2(1) with t = t givesΘ (V ⊕ φ) =Θ(V ⊕ φ) + ∆(co Noting that cocurv ∇ = − curv∇ and δ = ∆ (b stated results as a special case of Theorem 11.2 a Proof of Theorem 9.5. The hypothesis that g b So the generator equation defined in (2) above T M ⊕ h. The solution is unique because ∆ is inj 9.5 follows. The Cartan connection on g in Theor g (1) ; conclusion (3) above implies that it has the Suppose g is reductive and let ∇ be a norma some instead denote the isotrop in 5.8. Then it is not too difficult to check that this regard, a helpful formula, readily derived, is The generator ∇ (1) is necessarily the Cartan c in Proposition 5.8. Its curvature is given by In particular, curv ∇ (1) vanishes if and only if cur invariant. But as id T M is a section of T * M ⊗ curv ∇ = 0. In that case we obtain

Application: conforma
In this section we turn to the application of Ca tures. Our results are summarized in Theorems 1 12.1. The Lie algebroid setting. Let σ be a connected manifold M , with n := dim M 3. L viewed as the one-dimensional subbundle of Sym σ.
Let g σ ⊂ J 1 (T M ) denote the isotropy of σ ⊂ Let g ⊂ J 1 (T M ) denote the isotropy of σ ⊂ this means that the 1-jet of a vector field V at by skew(φ) := (φ − φ t ). These morphisms and class of σ.
Because the Levi-Cevita connection ∇ associa it also generates g ⊃ g σ .

Classical ingredients.
From well-known r we know that the curvature of the Levi-Cevita con Also of significance will be the Cotton-York ten Bianchi's second identity 6.5(4) for the genera tween the Cotton-York tensor d ∇ R, and the deriv known that W = 0 implies the vanishing of d ∇ R where E Weyl = 0 and the values of d ∇ R are restr 12.4. The W = 0 case. Our second theorem li algebroid language, results that are essentially cl Theorem. Suppose W = 0. Then g has an ass its prolongation g (1) ⊂ J 2 (T M ), which is surjec Cartan connection on g (1) by ∇ (2) , we have: conformal Killing fields. Outline of the application of Cartan's tial results for the general case W = 0, we sket results above. Although g ⊂ J 1 (T M ) is surjective, we hav not apply. In 12.7 we show that g is already Θ coboundary morphism is not injective and Theor We turn then, in 12.8 and 12.9, to the prolonga surjective (because g is Θ-reduced) but has non-tr We show in 12.10 that g (1) is already Θ-reduc morphism associated with g (1) is injective and Th it an associated Cartan algebroid.
12.6. The W = 0 case and intransitivity. I Θ-reduced. According to Proposition 12.10 belo for all vector fields V, U 1 , U 2 , U 3 . We shall see i this definition is independent the metric within Levi-Cevita connection ∇.
The remainder of this section is devoted to pr rems. 12.7. The torsion reduction of g. Since g ⊂ J as torsion reduction. To compute it, we turn to morphism for g, T * M ⊗ h ∆ − → Alt 2 (T M ) Its restriction to T * M ⊗ h σ is nothing but the up Since the latter is an isomorphism (see 9.6) the f H(g) = 0, implying g is already torsion-reduce however, because ∆ has non-trivial kernel. Indee (1) rank(ker ∆) = rank(T 12.8. The first prolongation g (1) . Since g is reduced) the prolongation g (1) is surjective (Prop its structure kernel h (1) is ker δ = ker ∆. Define a This formula may also be written 10. The Θ-reduction of g (1) . Since g (1) ⊂ J the Θ-reduction of g (1) is the kernel of a morphism by Θ (1) , to distinguish it from the corresponding 9.2. The definition of h(g (1) ) depends on the low which we denote by Note that the first term on the right belongs to A Alt 2 (T M )⊗ id T M . In particular, the image of δ ( h. Since coRicci is injective (n 3) we have ke prolongation g (2) := (g (1) ) (1) of g has trivial stru particular, h(g (1) ) := (Alt 2 (T M ) ⊗ g)/ im δ (1) ha Next, we observe that the composite morphism is injective. This follows from the description of where E Ricci = coRicci(Sy Identifying E Weyl with the corresponding g-subre Proposition. The following diagram commutes: Evidently, W is strongly degenerate if and only i Proof of proposition. We will apply part (2) of g, t, h, δ, Θ,Θ, g (1) in the theorem being played by Our first task is to choose a connection ∇ h (1) ∇ on T M determines a linear connection on T * M chain of inclusions The ∇-invaria from Proposition 6.2 (2). So the second inclusion generates g and because ∆ : T * M ⊗ h → Alt 2 (T M is g-equivariant, it follows that ∆ is∇-equivaria is torsion free, meaning∇-invariance is the sam h (1) ⊂ T * M ⊗ h of ∆ must be ∇-invariant, as cla We choose ∇ h (1) to be the connection that h ( invariant subbundle. Appealing to 12.9(1) and t T M , one can show that In the present context 11.2(1) reads for arbitrary sections X ⊂ g and φ ⊂ h (1) . S composite We have used (2) above. Referring to the descrip solution is given by ǫ = −X ·R. Using ∇ (1) to iden in mind the identification h (1) ∼ = T * M implicit ab U X + skew(α ⊗ U ) + α(U for arbitrary sections X ⊂ g and α ⊂ T * M . We claim Proof of (2). Since W = 0 we have curv ∇ = co one computes

Equation (
2) now follows from the readily verifie One also makes use of the fact that tor ∇ = 0. A.2. The flat Cartan algebroid associated G be a Lie pseudogroup of transformations in M pseudoaction F of some Lie groupoid G over M . each point g ∈ G lies in some bisection b ∈F same one-jet at g. Thus F defines a map D F : G all one-jets of bisections of G. This map, which projection J 1 G → G, is a groupoid morphism bec An arbitrary groupoid morphism D : G → J J 1 G → G is what we call a Cartan connection o viewed as certain 'multiplicatively closed' distrib is Frobenius integrable precisely when it comes in which case D is simply the tangent distributio a (possibly non-integrable) Cartan connection is groupoids are deformed Lie pseudogroups.
Differentiating a Cartan connection D : G → J for the exact sequence of Lie algebroids (1) 0 → T * M ⊗ g ֒→ J 1 g → foliation F is a pseudoaction generating a Lie pse M .
For each locally defined ∇-parallel section X integrates to a one-parameter family of local tran versely each transformation in the pseudogroup G 'close' to the identity -arises as the time-one m field. In this sense G integrates the flat Cartan a Appendix B. Misce B.1. On morphisms whose domains sit in a category of vector spaces, or of vector bundles arbitrary morphism, B 0 its kernel, and suppose B as shown below: The proof of the following is a straightforward di Lemma. Let A 0 and A 1 denote, respectively, the morphism A ֒→ B shows that the rank-r subbundle E h ⊂ E is g-in on E h , the representation g → gl(E h ) factors representation T M → gl(E h ), i.e., a flat linear c to be any non-vanishing D-parallel section of E h , flatness and the simple-connectivity of M . The u B.3. Proof of Proposition 10.3. Let ∇ be any and dθ are all g-invariant, they are all∇-invari ∇ -invariance of σ and n, one immediately compu (∇ U σ)(V 1 , V 2 ) = σ (tor∇(U (1) and ∇n = tor∇(n), (2) where tor∇(U ) := tor∇(U, · ) ⊂ T * M ⊗ T M a fields on M . Here and in the sequel a subscript sy indicates its symmetrization (resp. skew-symmetr σ as appropriate. For any 2-tensor φ, we have φ From (2) it follows that ∇ n n = 0. From (2 compute, θ(∇ V n) = θ(tor∇(n, V )) = dθ(n, V So ∇ V n is H-valued, for any V ⊂ T M . This esta Now Alt 2 (H) is rank-one and spanned by dA tor∇ to H (a section of Alt 2 (H)) is of the form Therefore, if ∇ is a generator satisfying ∇σ|H becomes a consequence of (4) above.
We return to supposing that ∇ is an arbitrary remaining claims of the proposition we require coboundary morphism, Relationship with the Levi-Cevita connection. so, it is not difficult to express the generator ∇ Levi-Cevita connection ∇ L-C associated with σ: Since J 2 t itself occurs in a natural exact sequenc 0 → Sym 2 (T M ) ⊗ t → J 2 t the bundles ker ω 2 and J 2 t have the same rank.
Recalling that X ⊂ J 1 t is holonomic if and on that X is holonomic if and only if J 1 X ⊂ ker ω 2 . Proposition 8.2.