INFINITESIMALLY NATURAL PRINCIPAL BUNDLES

. We extend the notion of a natural ﬁbre bundle by requiring dif-feomorphisms of the base to lift to automorphisms of the bundle only inﬁnites- imally, i.e. at the level of the Lie algebra of vector ﬁelds. We classify the principal ﬁbre bundles with this property. A version of the main result in this paper (theorem 4.4) can be found in Lecomte’s work [12]. Our approach was developed independently, uses the language of Lie algebroids, and can be generalized in several directions.

Natural bundles were introduced 1 by Nijenhuis [18,19], building on the theory of 'geometric objects' [24,29,17]. A classification theorem of Palais and Terng [20], completed by Epstein and Thurston [5] and based on earlier work of Salvioli [23], states that any natural fibre bundle is associated to the k th order frame bundle F k (M ). Definition 1.2. An infinitesimally natural principal fibre bundle is a principal fibre bundle π : P → M , together with a distinguished Lie algebra homomorphism σ : Γ c (T M ) → Γ c (T P ) G that splits the exact sequence of Lie algebras (2).
This extends the notion of a natural fibre bundle in two separate ways. First of all, we do not require locality, but prove it. And secondly, we only require diffeomorphisms of the base to lift to automorphisms of the bundle infinitesimally, i.e., at the level of Lie algebras.
The outline of the paper is as follows. Sections 2, 3 and 4 are devoted to the classification of infinitesimally natural principal fibre bundles. The central result is theorem 4.4, which states the following.
Theorem. Any infinitesimally natural principal fibre bundle is associated to the universal cover F +k (M ) of the connected component of the k th order frame bundle.
The number k is at most dim(G), unless dim(M ) = 1 and dim(G) = 2, in which case also k = 3 may occur.
A version of this result was obtained earlier by Lecomte [12], using the theory of distributions and foliations. In the present paper, we use instead the theory of Lie algebroids and groupoids, which opens the door to generalizations in a context where additional structure or symmetry is present on the underlying manifold.
We extend Theorem 4.4 to fibre bundles with a finite dimensional structure group in Section 5, with special attention for vector bundles. Finally, in Section 6, we exhibit conditions under which a splitting of (2) gives rise to a flat connection.
2. Principal bundles as Lie algebra extensions. We seek to classify infinitesimally natural principal fibre bundles. It is to this end that we study Lie algebra homomorphisms σ that split (2). In this section, we will prove that σ must be a differential operator of finite order.
The first step, to be taken in Section 2.1, is to show that maximal ideals in Γ c (T M ) correspond precisely to points in M . Using this, we will prove in Section 2.2 that σ must be a local map. We will then prove, in Section 2.3, that σ is in fact a differential operator of finite order.
2.1. Ideals of the Lie algebra of vector fields. The following proposition, due to Shanks and Pursell [25], constitutes the linchpin of the proof. It identifies the maximal ideals of the Lie algebra Γ c (T M ) of smooth, compactly supported vector fields on M . The proof is taken from [25], with some minor clarifications.
For q ∈ M , define I q ⊆ Γ c (T M ) to be the ideal of vector fields v ∈ Γ c (T M ) which are zero and flat at q, that is, v(q) = 0 and (ad(w i1 ) . . . ad(w in )v)(q) = 0 for all w i1 , . . . w in , ∈ Γ c (T M ). To prove Proposition 1, we need the following, somewhat technical result.
Lemma 2.1. Suppose that a maximal ideal I of Γ c (T M ) contains a vector field which does not vanish at q ∈ M . Then q has a neighbourhood U q such that for every w ∈ Γ c (T M ) with Supp(w) ⊂ U q , there exist vector fields v ∈ I and u ∈ Γ c (T M ) such that w = [v, u].
Although one can easily find u ∈ Γ c (T M ) and v ∈ I such that w = [u, v] locally, it is not always clear how to extend these to global vector fields while simultaneously satisfying w = [u, v]. For instance, with M the circle S 1 and v = w = ∂ θ , the solution u = θ∂ θ does not globally exist. We need to do some work in order to define a proper cutoff procedure.

Proof.
Choose v ∈ I with v(q) = 0. There exist local co-ordinates x 1 , . . . , x n and an open neighbourhood W of q such that v| W = ∂ 1 . Choose W to be a block centred around q, and nest two smaller blocks (in local co-ordinates) inside, so that W ⊃ V ⊃ U .
For i = 1, the i th component of the above reads h∂ 1 u i = w i . In the region x 1 ≤ 1 3 ε, we set u i (x 1 , . . . , x n ) = x1 −∞ w i (t, x 2 . . . , x n )dt. On U , whereṽ = ∂ 1 , we then have ∂ 1 u i = w i and therefore h∂ 1 u i = w i . This is obviously also correct for points outside U with x 1 ≤ 1 3 ε, as both u i and w i are zero. For 1 3 ε ≤ x 1 ≤ 2 3 ε, where w = 0, we let u i (x 1 , x 2 , . . . , x n ) be a constant function of x 1 , so that u i (x 1 , x 2 , . . . , x n ) = u i ( 1 3 ε, x 2 , . . . , x n ). We then have ∂ 1 u i (x 1 , . . . , x n ) = 0, guaranteeing h∂ 1 u i = w i . Note that this does not effect the smoothness of u i .
Finally, for 2 3 ε ≤ x 1 ≤ ε, let u i tend to zero, and let u i be zero for x 1 ≥ ε. This can be done in such a way that u i remains smooth. Since both h and w are zero, we have h∂ 1 u i = w i on all of M .
The case i = 1 is handled similarly, the only difference being that the first . , x n ) in order for h∂ 1 u 1 − u 1 ∂ 1 h = w 1 to hold. Since this renders u 1 zero on a neighbourhood of the boundary of V , one is then free to define u 1 to be zero on M \V .
Thus, for every vector field w with support in U , we have constructed vector fieldsṽ ∈ I and u ∈ Γ c (T M ) such that w = [ṽ, u].
Using Lemma 2.1, one readily proves the following. Lemma 2.2. Suppose that I is an ideal such that for all q ∈ M , there exists a v ∈ I with v(q) = 0. Then I = Γ c (T M ).
Proof. Let w ∈ Γ c (T M ). Cover the support Supp(w) of w by finitely many neighbourhoods U q1 . . . U q N with the properties described in Lemma 2.1. Using a partition of unity, write w = N j=1 w j with Supp(w j ) ⊆ U qj . By Lemma 2.1, there exist v j ∈ I and u j ∈ Γ c (T M ) such that w j = [v j , u j ]. In particular, every w j is an element of I. It follows that also w = N j=1 w j is in I.
Using Lemma 2.2, we now prove Proposition 1.
Proof of Proposition 1. For every proper ideal I ⊂ Γ c (T M ), there exists a q ∈ M such that I ⊆ I q . Indeed, Lemma 2.2 guarantees the existence of a q ∈ M such that v(q) = 0 for all v ∈ I. Since ad(w 1 ) . . . ad(w n )v ∈ I for all v ∈ I and w 1 , . . . , w n ∈ Γ c (T M ), it follows that also (ad(w 1 ) . . . ad(w n )v)(q) = 0. Every v ∈ I is therefore not only zero at q, but also flat. We conclude that I ⊆ I q .
It remains to show that for every q ∈ M , the ideal I q is indeed maximal. Suppose that I q ⊆ I for a proper ideal I ⊂ Γ c (T M ). Then there exists a pointq ∈ M such that I ⊆ Iq, and hence I q ⊆ Iq. It follows thatq = q. Since I q ⊆ I ⊆ Iq, it follows that I = I q , so that the ideal I q is maximal.
A maximal subalgebra A of a Lie algebra L is either self-normalizing or ideal. Indeed it is contained in its normaliser, which therefore equals either A or L. A theorem of Barnes [2] states that a finite-dimensional Lie algebra is nilpotent if and only if 2 every maximal subalgebra is an ideal. On the other extreme: Proposition 2. Let L be a Lie algebra over a field K, and let S be the set of subspaces A ⊂ L such that A is both an ideal and a maximal subalgebra. Then L where A has codimension 1 in L, and X / ∈ A. Then A is an ideal maximal subalgebra, which does not contain X.
Let A be an ideal maximal subalgebra, and X / ∈ A. Then A+KX is a subalgebra strictly containing A, so that it must equal L.
As a corollary, we have the following well known statement (cf. e.g. [1,Thm. 1.4.3]) Proof. According to lemma 1, the maximal ideals are precisely the ideals I q of vector fields in Γ c (T M ) which are zero and flat at q. I q is strictly contained in the subalgebra A q of vector fields which are zero at q, so that no ideal is a maximal subalgebra. So every maximal subalgebra is self-normalizing, and the result follows from Proposition (2).

2.2.
The splitting as a local map. With the main technical obstacles out of the way, we turn our attention to the sequence (2). We now prove that σ is a local map.
Recall that the Lie algebra Γ c (T P ) G v of vertical, G-invariant vector fields on P with support in the preimage of a compact subset of M is identified with the Lie algebra Γ c (ad(P )) of compactly supported sections of the adjoint Lie algebra bundle ad(P ). Concretely, the section m → [p m , X] of ad(P ) := P × ad g is identified with the vertical, G-invariant vector field v p = d dt | 0 p m e tX on P . Lemma 2.3. Let P → M be a principal G-bundle over M , with G any Lie group. Let σ : Γ c (T M ) → Γ c (T P ) G be a Lie algebra homomorphism splitting the exact sequence of Lie algebras Then σ is local in the sense that π(Supp(σ(v))) ⊆ Supp(v). We identify Γ c (T P ) G with Γ c (T P/G), the compactly supported sections of the vector bundle T P/G → M that arises as the quotient of T P by the pushforward of the right G-action on P . This allows us to consider the splitting σ as a map σ : Γ c (T M ) → Γ c (T P/G). Since σ is local by Lemma 2.3, it defines a morphism from the sheaf of smooth sections of T M → M to the sheaf of smooth sections of T P/G → M . By Peetre's Theorem ( [21]), the map σ : Γ c (T M ) → Γ c (T P/G) must be a differential operator of locally finite order.

Proof. Let
2.3. The splitting as a differential operator. In this section, we will prove that σ is a differential operator of finite order. Since we already know that it is of locally finite order, it remains to find a global bound on the order.
Recall that A m ⊆ Γ c (T M ) is the subalgebra of vector fields that vanish at m, andσ m : A m → g is the restriction of σ to A m , followed by the map Γ(ad(P )) → g that picks out the fibre over m and identifies it with g.
The fact that σ is a differential operator of locally finite order implies that for each m ∈ M , the Lie algebra homomorphismσ m : A m → g factors through the jet Lie algebra J r,0 m (T M ) := A m /H r m , with H r m = {v ∈ A m | j r m (v) = 0} the ideal of vector fields that vanish up to order r.
Local co-ordinates provide one with a basis x α ∂ i of J r,0 m (T M ), where x α is shorthand for x α1 1 · · · x αn n . With Vec k n = Span{x α ∂ i | | α| = k + 1 , i = 1 . . . n}, we have Vec k n . Since we do not know r, we define Vec k n , and remark thatσ m induces a Lie algebra homomorphism Vec n → g. The Lie algebra Vec n depends on M only through its dimension n. Note that Vec k n is the k-eigenspace of the Euler vector field E := n i=1 x i ∂ i , and that each element of Vec n can be uniquely written as a finite sum of homogeneous vector fields. If I is an ideal containing v = N k=0 v k , then ad(E) j v = N k=0 k j v k ∈ I for all j. By taking suitable linear combinations, one sees that v k ∈ I. Thus any ideal splits into homogeneous components n . This renders the ideal structure of Vec n more or less tractable, enabling us to prove the following bound on the order ofσ m .
Lemma 2.4. The order of the differential operator σ is at most dim(g) unless dim(M ) = 1 and dim(g) = 2, in which case the order is at most 3.
Proof. We closely follow Epstein and Thurston [5]. One checks by hand that the only ideals of the Lie algebra Consider Vec 1 as a subalgebra of Vec n , define K to be the kernel of the homomorphism Vec n → g induced byσ m , and let K 1 := K ∩Vec 1 . We then have injective homomorphisms Vec 1 /K 1 → Vec n /K → g , so that dim(Vec 1 /K 1 ) ≤ dim(g). As K 1 is an ideal, it must be of the form mentioned above. This leads us to conclude that x k 1 ∂ 1 ∈ K for all k > dim(g) unless dim(g) = 2, in which case x k 1 ∂ 1 ∈ K for all k > 3, and x 2 1 ∂ 1 ∈ K. The following short calculation shows that if dim(g) = 2 and dim(M ) > 1, then also x 3 1 ∂ 1 ∈ K. As K contains But by bracketing with x 2 1 ∂ 2 and x 2 ∂ 1 respectively, we see that to the effect of replacing x 1 by x i up to a nonzero factor. This shows that In the generic case dim(g) = 2, dim(M ) = 1, we may conclude that the order of σ is at most dim(g), because H dim(g) m ⊂ K. In the exceptional case that dim(g) = 2 and dim(M ) = 1, the order of σ is at most 3.
In particular, σ is a differential operator of finite rather than locally finite order. We summarise our progress so far in the following proposition.
The point is that although σ is defined only on sections, ∇ comes from a veritable bundle map J k (T M ) → T P/G.

2.4.
The compact support condition. We defined infinitesimally natural principal fibre bundles in terms of a splitting σ : Γ c (T M ) → Γ c (T P ) G of the sequence (2) of compactly supported sections. We now see that, equivalently, an infinitesimally natural principal fibre bundle can be defined as a bundle with a continuous splitting τ : Γ(T M ) → Γ(T P ) G of the exact sequence of Fréchet Lie algebras (with the usual topology of uniform convergence of all derivatives on compact subsets, the non-compactly supported version of (2). Indeed, the extensionσ : Γ(T M ) → Γ(T P ) G of σ is a splitting of (4), which is local (in the sense that π(supp(σ(v))) ⊆ supp(v)) and continuous because it is a differential operator. Conversely, one sees from the proof of Lemma 2.3 that every Lie algebra homomorphism τ : and hence restricts to a splitting of (2). If, furthermore, τ : Γ(T M ) → Γ(T P ) G is continuous, then it is uniquely determined by its restriction to the dense subalgebra Γ c (T M ), and hence local. For later use, we formulate this observation as a proposition.
Proposition 4. Every splitting of (2) induces a continuous splitting of (4). Conversely, every splitting of (4), continuous or not, induces a splitting of (2). Consequently, an infinitesimally natural principal fibre bundle can be alternatively characterised as a principal fibre bundle P , together with a distinguished continuous Lie algebra homomorphism τ : Γ(T M ) → Γ(T P ) G that splits the exact sequence of Fréchet Lie algebras (4).

3.
Lie groupoids and algebroids of jets. The bundles J k (T M ) and T P/G are Lie algebroids, and it will be essential for us to prove that ∇ : J k (T M ) → T P/G is a homomorphism of Lie algebroids. In order to do this, we will first have a closer look at J k (T M ) and T P/G, and at their corresponding Lie groupoids.
Let us first set some notation. The jet group G k 0,0 (R n ) is the group of k-jets of diffeomorphisms of R n that fix 0. It is the semi-direct product of GL(R n ) and the connected, simply connected, unipotent Lie group of k-jets that equal the identity to first order.
The subgroup G +k 0,0 (R n ) of orientation preserving k-jets is connected, but not simply connected. As G +k 0,0 (R n ) retracts to SO(R n ), its homotopy group is isomorphic to {1} if n = 1, to Z if n = 2, and to Z/2Z if n > 2. For brevity, we introduce the following notation.

3.1.
The Lie groupoid of k-jets. In this section, we define the Lie groupoid G k (M ) of k-jets, its maximal source-connected Lie subgroupoid G +k (M ), and the k th order frame bundle F k (M ).
Denote by G k m ,m (M ) the manifold of k-jets at m of diffeomorphisms of M which map m to m , and denote by  Proof. We may as well consider k = 1, because the fibres of is isomorphic to the frame bundle. By definition, M is oriented precisely when the frames can be grouped into positively and negatively oriented ones. Definition 3.3. We define G +k (M ) to be the maximal source-connected Lie subgroupoid of G k (M ), and denote its source fibre by F +k (M ).
In the light of the previous lemma, this means that G +k (M ) is the Lie groupoid of k-jets of orientation preserving diffeomorphisms if M is orientable, and simply is a homomorphism of groups. We will call it the k th order derivative. Because Dα is source-preserving and right invariant, it defines a homomorphism Diff(M ) → Aut G k m,m (M ) (G k * ,m (M )), splitting the exact sequence of groups (1). This makes 3.2. The Lie algebroid of k-jets. The bundle J k (T M ) possesses a structure of Lie algebroid, induced by the Lie groupoid G k (M ). We now describe the Lie bracket on Γ(J k (T M )) explicitly. Later, in section 3.4, we will use this to show that ∇ is a Lie algebroid homomorphism.
The anchor dt : J k (T M ) → T M is easily seen to be the canonical projection, so we shall denote it by π. The Lie bracket on Γ(J k (T M )) however, which is defined as the restriction of the commutator bracket on Γ(T G k (M )) to the right invariant source preserving vector fields, perhaps deserves some comment.
Define J k,0 (T M ) to be the kernel of π, and consider the exact sequence of Lie algebras  Proof. The first equality is clear, as j k is a homomorphism of Lie algebras. The second equality can be seen as follows. Consider the bundle of groups G k (M ) * , * := {j k m (α) ∈ G k (M ) | α(m) = m}, with bundle map s = t. Its sections Γ(G k (M ) * , * ) form a group under pointwise multiplication, whose Lie algebra is Γ(J k,0 (T M )) with the pointwise bracket. As Γ(G k (M ) * , * ) acts from the left on G k (M ) by j k m (γ) → j k m (α m ) • j k m (γ), respecting both the source map and right multiplication, the inclusion Γ(J k,0 (T M )) → Γ(J k (T M )) is a homomorphism of Lie algebras. This proves the second line.
To verify the third line, we must choose a smooth map   , p)]. An element [(q, p)] corresponds precisely to a G-equivariant diffeomorphism π −1 (p) → π −1 (q), and the product to composition of maps. Its Lie algebroid T P/G is called the Atiyah algebroid. Indeed, the subspace of T idm ((P × P )/G) which annihilates ds is canonically (T P/G) m . A section of T P/G can be identified with a G-equivariant section of T P , endowing Γ(T P/G) with the Lie bracket that comes from Γ(T P ) G .
3.4. The splitting as a homomorphism of Lie algebroids. The point of considering the Lie algebroid structure of J k (T M ) was of course to prove the following. Proof. The fact that ∇ respects the anchor is immediate.
We now show that ∇ : Secondly, its restriction to Γ(J k,0 (T M )) is a homomorphism. If τ x = j k x (v x ) and υ x = j k x (w x ) are sections of J k,0 (T M ), then ∇τ and ∇υ are in the kernel of the anchor. This implies that their commutator at a certain point m depends only on their values at m, not on their derivatives. To find the commutator at m, we may therefore replace j k We already know that this is The last step is to show that ∇ respects the bracket between j k (Γ(T M )) and Γ(J k,0 (T M )). Again, let j k (v) be an element of the former and τ x = j k x (w x ) of the latter. Considered as an equivariant vector field on P , the vertical vector field ∇(τ ) takes the value σ p (w π(p) ) at p ∈ P . Then [∇(j k (v)), ∇(τ )] is the Lie derivative along σ(v) of the vertical vector field σ p (w π(p) ). Differentiating along σ(v) is done by considering (p, p ) → σ p (w π(p ) ), differentiating w.r.t. p and p separately, and then putting p = p . This results in , τ ]) as required. Therefore ∇ must be a homomorphism on all of Γ(J k (T M )).
Definition 3.5. A connection ∇ of a Lie algebroid A on a vector bundle E is by definition a bundle map of A into DO 1 (E), the first order differential operators on E, which respects the anchor, cf. [13,§ III.5]. If moreover it is a morphism of Lie algebroids, then the connection is called flat. A flat connection of A on E is also called a representation of A on E.
This explains our notation for the map ∇ induced by σ. Given a representation V of G, one may form the associated vector bundle E := P × G V . The map ∇ then defines a Lie algebroid homomorphism of Γ(J k (T M )) into the Lie algebroid of first order differential operators on E. (Simply consider a section of E as a G-equivariant function P → V , and let Γ G (T P ) act by Lie derivative.) By definition, this is a flat connection, or equivalently a Lie algebroid representation.
3.5. Infinitesimally natural transitive Lie algebroids. The above is readily generalized to transitive Lie algebroids L → M , whose anchor ρ : L → T M is surjective. The kernel K ⊆ L of ρ is then a Lie algebroid K → M with trivial anchor. In particular, every fibre K m has a Lie algebra structure. The anchor gives rise to the exact sequence of Lie algebras The order k of the differential operator σ satisfies k ≤ rank(K), except when n = 1 and rank(K) = 2, in which case k ≤ 3.

4.
The classification theorem. We use the fact that ∇ : J k (T M ) → T P/G is a homomorphism of Lie algebroids to find a corresponding homomorphism of Lie groupoids. This will give us the desired classification of infinitesimally natural principal fibre bundles.

4.1.
Integrating a homomorphism of Lie algebroids. The following theorem states that homomorphisms of Lie algebroids induce homomorphisms of Lie groupoids if the initial groupoid is source-simply connected. Theorem 4.1 (Lie II for algebroids). Let G and H be Lie groupoids, with corresponding Lie algebroids A and B respectively. Let ∇ : A → B be a homomorphism of Lie algebroids. If G is source-simply connected, then there exists a unique homomorphism G → H of Lie groupoids which integrates ∇.
Remark. The result was probably announced first in [22], and proofs have appeared e.g. in [16] and [14]. We follow [4], which the reader may consult for details.
Sketch of proof. The idea is that ∇ allows one to lift piecewise smooth paths of constant source in G to piecewise smooth paths of constant source in H. Sourcepreserving piecewise smooth homotopies in G of course do not affect the endpoint of the path in H, so that, if G is source-simply connected, one obtains a map G → H by identifying elements g of G with equivalence classes of source preserving paths from id s(g) to g. One checks that this is the unique homomorphism of Lie groupoids integrating ∇.
Unfortunately, G k (M ) is not always source connected, let alone source-simply connected. Recall that G +k (M ) is the maximal source-connected Lie subgroupoid of G k (M ), and therefore has the same Lie algebroid J k (T M ).
We defineG +k (M ) to be the set of piecewise smooth, source preserving paths in G +k (M ) beginning at an identity, modulo piecewise smooth, source preserving homotopies. It is a smooth manifold because G +k (M ) is, and a Lie groupoid under the unique structure making the projection on the endpointG +k (M ) → G +k (M ) into a morphism of groupoids. Explicitly, the multiplication is given as follows. If g(t) a path from id m to g(1) m m , and h(t) a path from id m to h(1) m m , then the product [h] • [g] is [(h · g(1)) * g], where the dot denotes groupoid multiplication and the star concatenation of paths. The proof of associativity is the usual one.
Note that the source fibreG +k (M ) * ,m is precisely the universal cover of the connected component of the k th order frame bundle G +k (M ) * ,m = F +k (M ). In order to cut down on the subscripts, we introduce new notation forG +k (M ) * ,m and its structure groupG +k (M ) m,m . Definition 4.2. We denote the universal cover of the connected component of the k th order frame bundle by F +k (M ), and its structure group by G(k, M ).
It is an infinitesimally natural bundle because F +k (M ) is. Note that G(k, M ) is not the universal cover of G +k m,m (M ), but rather its extension by π 1 (F k (M )). As π 1 (F k (M )) = π 1 (F (M )), we have the exact sequence of groups The group G +k m,m (M ) in turn is isomorphic to G +k 0,0 (R n ) if M is orientable, and to G k 0,0 (R n ) if it is not.

4.2.
Classification. Now that we have found a source-simply connected Lie groupoid with J k (T M ) as Lie algebroid, we can finally apply Lie's second theorem for algebroids to obtain the following.
Proposition 6. If σ splits the exact sequence of Lie algebras (2), then it induces a morphism of groupoids exp ∇ :G +k → (P × P )/G such that the following diagram commutes, with exp m the flow along a vector field starting at id m .

Γ(T M ) Γ(T P/G)
Proof. AsG +k (M ) is a source-simply connected Lie groupoid with J k (T M ) as Lie algebroid, we can apply Lie's second theorem for algebroids.
It is perhaps worthwhile to formulate this for general transitive Lie groupoids, as it clarifies the link with the recent work of Grabowski, Kotov and Poncin [8]. For the Atiyah algebroid A of a principal fibre bundle with a connected, reductive structure group, they classify Lie algebra isomorphisms of Γ(A) in terms of the Lie algebroid isomorphisms of A. Sketch of proof. Using Theorem 3.7, this is analogous to the case of the gauge groupoid.
We have paved the way for a classification of infinitesimally natural principal fibre bundles.
Theorem 4.4. Let π : P → M be an infinitesimally natural principal G-bundle with a splitting σ of (2). Then there exists a group homomorphism ρ : G(k, M ) → G such that the bundle P is associated to F +k (M ) through ρ, i.e.
Moreover, σ is induced by the canonical splitting for F +k (M ). It is also surjective and G-equivariant, and hence an isomorphism of principal G-bundles. As ((P × P )/G) * ,m P andG +k * ,m (M ) × ρ ((P × P )/G) m,m F +k (M ) × ρ G, the equivalence is proven. The remark on σ follows from the construction.
This classifies the infinitesimally natural principal fibre bundles. They are all associated (via a group homomorphism) to the bundleG +k * ,m = F +k (M ). The classification of natural principal fibre bundles is now an easy corollary. The following well known result ( [20,28]) states that they are precisely the ones associated to G k * ,m (M ) = F k (M ).

BAS JANSSENS
Corollary 2. Let π : P → M be a natural principal G-bundle with local splitting Σ of (1). Then P is associated to F k (M ). That is, there exists a homomorphism ρ : The Lie algebra homomorphism σ : Γ c (T M ) → T P/G defined by σ(v) := ∂ t | 0 Σ(exp(tv)) is local by assumption, and according to proposition 3 it factors through the k-jets for some k > 0. It suffices to show that Σ(φ) m,m = id m,m for any φ ∈ Germ m,m (M ) that agrees with the identity to k th order at m.
In local co-ordinates where v : R n → R n vanishes to k th order. We define the one parameter family of germs of diffeomorphisms , the image of a vector field that vanishes to order k at m. Therefore t → Σ(φ t ) m,m is constant, and Σ(φ) m,m = id m,m as required.
To summarise: natural principal fibre bundles are associated to a higher frame bundle, whereas infinitesimally natural principal fibre bundles are associated to the universal cover of a higher frame bundle.
The kernel of the corresponding cover of groups is precisely i * π 1 (G +k m,m (M )), with i : G +k m,m (M ) → G +k * ,m (M ) the inclusion. Note that i * has a nonzero kernel precisely when a vertical loop is contractible in G +k * ,m (M ), but not by a homotopy which stays inside the fibre. Denoting π 1 (G +k m,m (M )) by Z, we obtain the exact sequence A moment's thought reveals that this extension is central: if g(t) is a path in G +k m,m (M ) and h(t) one in G +k * ,m (M ), then both h * (i • g) and (i • g) · h(1) * h can be homotoped into t → h(t)g(t).
We may as well restrict attention to the case k = 1, in which

4.3.2.
Orientable manifolds. For orientable manifolds, the situation simplifies. If we identify the connected component of G 1 m,m (M ) with GL + (R n ), we obtain a homomorphism i * of GL + (R n ) intoG 1,+ m,m (M ). There is a second homomorphism π 1 (F (M )) →G 1,+ m,m (M ). Their images intersect in Z/Ker(i * ), and commute by an argument similar to the one on centrality of (7). The group ( GL + (R n )×π 1 (F (M ))) Z is defined as the quotient of GL + (R n ) × π 1 (F (M )) by the equivalence relation (gz, h) ∼ (g, zh), and can be regarded as a subgroup ofG 1,+ m,m (M ). Note that if Ker(i * ) is nonzero, the above equivalence relation sets it to 1.
If M is orientable, we may restrict our attention to F + (M ), which then has connected fibres. Any path in F + (M ) which starts and ends in the same fibre can therefore be obtained by combining a closed loop with a path in GL + (R n ). For orientable manifolds, we thus haveG 1,+ m,m (M ) ( GL + (R n ) × π 1 (F + (M ))) Z , and in the same vein Remark. Note that an infinitesimally natural bundle P is natural, i.e. associated to F k (M ) rather than F k (M ), if and only if the group homomorphism ρ : G(k, M ) → G of Theorem 4.4 is trivial on π 1 (F + (M )).

Spin manifolds.
Let M be an orientable manifold, equipped with a pseudo-Riemannian metric g of signature η ∈ Bil(R n ). Then OF + g := {f ∈ F + (M ) | f * g = η} is the bundle of positively oriented orthogonal frames. A spin structure is then by definition an SO(η)-bundle 4 Q over M , plus a map u : Q → OF + g such that the following diagram commutes, with κ the canonical homomorphism SO(η) → SO(η).

SO(η)
SO(η) A manifold is called spin if it admits a spin structure. DefineQ := Q× SO(η) GL + (R n ), and let us again denote the induced mapQ → F + (M ) by u. As any cover of F + (M ) by a GL + (R n )-bundle can be obtained in this way, there is a 1:1 correspondence between spin covers of OF + g (M ) and F + (M ). In particular, whether or not M is spin does not depend on the metric.
The Serre homotopy exact sequence gives rise to the exact sequence The following proposition is well known.

Proposition 7.
A spin structure exists if and only if i * : Z → π 1 (F + (M )) is injective and (9) splits as a sequence of groups. If spin structures exist, then equivalence classes of spin covers correspond to splittings of (9).
Proof. See for example [15]. Our criterion for M to be spin is equivalent to the vanishing of the second Stiefel-Whitney class, see e.g. [11].
Remark. In terms of group cohomology, one can consider the sequence (9)  5. More general fibre bundles. In this section, we will prove a version of theorem 4.4 for fibre bundles which are not principal. It would however be overly optimistic to expect an analogue of of theorem 4.4 to hold for arbitrary smooth fibre bundles, so we will restrict ourselves to those bundles that carry a sufficiently rigid structure on their fibres, such as vector bundles. 5.1. Structured fibre bundles. We start by making this statement more precise. The following definition is modelled after the definition of a vector bundle, to which it reduces in the case that C is the category of vector spaces.
Definition 5.1. Let C be a category with a faithful functor F to the category of manifolds. Let C 0 be an object in C such that Aut(C 0 ) is a finite dimensional Lie group, and F induces a smooth action on F 0 := F(C 0 ).
Then a 'structured fibre bundle' with fibre C 0 is a smooth fibre bundle π : F → M with generic fibre F 0 = F(C 0 ), and for each m ∈ M a choice of structure C m ∈ ob(C) such that F(C m ) = π −1 (m). We also require that for each m ∈ M , there exist a trivialisation φ : π −1 (U ) → F 0 × U over some neighbourhood U of m which is structure preserving in the sense that for each x ∈ U , there exists an isomorphismφ x : C x → C 0 such that F(φ x ) = φ| π −1 (x) . Every vector bundle is associated to its frame bundle. We generalise this to structured fibre bundles.
Proposition 8. Let π : F → M be a structured fibre bundle with fibre C 0 . Then Fr(F ) := x∈M Iso(C 0 , C x ) is a smooth principal fibre bundle over M with structure group Aut(C 0 ), and F is isomorphic to Fr(E) × Aut(C0) F 0 as a smooth fibre bundle.
Proof. If φ is a local trivialisation of F over U , then the bijection φ : x∈U . We need to prove that for any two local trivialisations φ and ψ over U and V respectively, the equivariant bijection is a smooth map, or equivalently that the mapγ : U ∩ V → Aut(C 0 ) defined bŷ γ(x) := pr 2ψφ −1 (x, id) is smooth. Once we have established this, smoothness of φ•ψ −1 will imply thatψ •φ −1 is a diffeomorphism, that Fr(F ) is a smooth principal fibre bundle, and that the map The fact thatγ is continuous w.r.t. the topology of pointwise convergence on Aut(C 0 ) (induced by its action on F 0 ) follows directly from the fact that ψ • φ −1 : The action of Aut(C 0 ) on F 0 is smooth by assumption, and effective because F is faithful. We can therefore choose (f 1 , . . . , f k ) ∈ F 0 k with k = dim(Aut(C 0 )) such that the map Lie(Aut(C 0 )) → T f1 F 0 × . . . × T f k F 0 is injective. On a neighbourhood N ⊂ Aut(C 0 ) of the identity, the map A : Aut(C 0 ) → F k 0 : α → (α(f 1 ), . . . , α(f k )) is thus a diffeomorphism onto its image, and if R β is right multiplication by β ∈ Aut(C 0 ), the same goes for A • R β : R β −1 (N ) → A(N ). Sinceγ is continuous, we can choose W ⊂ U ∩ V and β ∈ Aut(C 0 ) such thatγ(x) ∈ R β −1 (N ) for all x ∈ W . Since the map A • R β •γ : W → A(N ) is given by it is certainly smooth. Since A • β −1 is a diffeomorphism,γ is smooth as well.
If π : F → M is any smooth fibre bundle, then an automorphism of π is by definition a diffeomorphism α of F such that π(f ) = π(f ) implies π(α(f )) = π(α(f )). It is called vertical if it maps each fibre to itself.
Definition 5.2. An automorphism of a structured fibre bundle F → M is an automorphism of the smooth bundle such that for every m ∈ M , there exists an isomorphismα m : C m → C m such that F(α m ) = α| π −1 (m) . The group of automorphisms is denoted Aut C (F ).
One can then construct a sequence of groups and its corresponding exact sequence of Lie algebras where 'V ' is for vertical, 'P ' for projectable, and c again stands for '0 outside a compact subset of M '. The proof of the following corollary of theorem 4.4 is now a formality.
Corollary 3. Let π : F → M be a structured fibre bundle with fibre C 0 such that (11) splits as a sequence of Lie algebras. Then there exists a homomorphism ρ : G(k, M ) → Aut(C 0 ) such that Proof. Using proposition 8, one constructs a natural isomorphism from Aut C (F ) to Aut(Fr(F )) under which the vertical subgroups of the two correspond, so that the exact sequence (10) is isomorphic to (1), and therefore (11) to (2). In view of the isomorphism F Fr(F ) × Aut(C0) F 0 , we can apply theorem 4.4 to Fr(F ) in order to substantiate our claim.

Vector bundles.
We specialise to the case of vector bundles. These are precisely structured fibre bundles in the category of finite dimensional vector spaces.
The exact sequence of Lie algebras (11) for a vector bundle E with fibre V is then where DO 1 c (E) is the Lie algebra of compactly supported 1 st order differential operators on Γ(E), and DO 0 c (E) is the ideal of 0 th order ones, that is to say Corollary 3 then says that (12) splits as a sequence of Lie algebras if and only if there is a representation ρ of G(k, M ) on V such that E F +k (M ) × ρ V .
But thanks to the fact that all finite dimensional representations of the universal cover of GL + (R n ) factor through GL + (R n ) itself, we can even say something slightly stronger.
Proposition 9. Let E → M be a vector bundle for which (12) splits as a sequence of Lie algebras. Then there exists a representation ρ of the group (G k × π 1 (M )) Pr Remark. If M is orientable, this reads E π * F +k (M ) × ρ V . In this expression, π * F +k (M ) is the pullback of F +k (M ) along π :M → M , considered as a principal G +k 0,0 (R n ) × π 1 (M )-bundle over M .
Proof. Consider the restriction of the map τ in equation (6) to the groupG +k m,m (M ). In order to prove the proposition, we need but show that its kernel Z acts trivially on V . For k = 0, this is clear.
If k is at least 1, the homomorphism GL + (R n ) →G +k m,m (M ) makes V into a finite dimensional representation space for GL + (R n ). But it is known (see [10, p. 311]) that all finite dimensional representations of its cover factor through GL + (R n ) itself. This implies that the subgroup Z which covers the identity must act trivially on V , and we may considerG +k * ,m (M )/Z (G k (M ) × π 1 (M )) Pr * ,m to be the underlying bundle, as announced.
This reduces the problem of classifying vector bundles with split sequence (12) to the representation theory of (G k × π 1 (M )) Pr m,m . The above extends a result [28] of Terng, in which she classifies vector bundles which allow for a local splitting of the sequence of groups (10). It is an extension first of all in the sense that we prove, rather than assume, that the splitting is local. Secondly, we have shown that in classifying vector bundles with split sequence (12) of Lie algebras rather than groups, one encounters only slightly more. Intuitively speaking, the extra bit is the representation theory of π 1 (M ). We refer to [28] for a thorough exposition of the representation theory of G k 0,0 (R n ).
6. Flat connections. In this section, we investigate splittings that come from a flat equivariant connection on a principal G-bundle P → M . We will prove that if the Lie algebra g of G does not contain sl(R n ) as a subalgebra, then the sequence of Lie algebras (2) splits if and only if P admits a flat equivariant connection. In other words, the sequence (2) then splits as a sequence of Lie algebras if and only if it splits as a sequence of Lie algebras and C ∞ (M )-modules. Note that this is certainly not the case for general groups G. The frame bundle for example always allows for a splitting of (2), but usually not for a flat connection.
Remark. For general Lie groups G, Lecomte has shown ([12, Thm. 3.1]) that the existence of a splitting is obstructed by the image of the Chern-Weil homomorphism under the map from de Rham cohomology to the Lie algebra cohomology of Γ(T M ) with values in C ∞ (M ). Combined with results of Shiga-Tsujishita [26], this implies that the characteristic classes of P → M are contained in the ideal generated by the Pontrjagin classes of M . 6.1. Lie algebras that do not contain sl(R n ). Although lemma 2.4 exhibits σ as a differential operator of finite order, the bound on the order is certainly not optimal. With full knowledge of the Lie algebras at hand, sharper restrictions can be put on the kernel of σ. In particular, if g does not contain sl(R n ), there is only a single relevant ideal, and σ is of order at most 1. For notation, see section 2.2. Lemma 6.1. Let n = 1, and let g be such that it does not contain two nonzero elements such that [X, Y ] = Y . Or let n ≥ 2, and let g be such that it does not admit sl(R n ) as a subalgebra. Then the kernel of the homomorphismf m : Vec n → g contains {v ∈ Vec n | Div m (v) = 0}.
Proof. We start with the case n = 1. Again, we note that the only ideals of Vec 1 = Span{x k ∂ x | k ≥ 1} are Span{x 2 ∂ x , x k ∂ x | k ≥ 4}, and for each N ≥ 1 an ideal Span{x k ∂ x | k ≥ N }. The corresponding quotients all contain two elements X and Y with [X, Y ] = Y , except the ideals corresponding to N = 1, 2. This means that also g, containing Vec 1 / ker(f m ) as the image off m , will possess X and Y such that [X, Y ] = Y unless the kernel off m contains the ideal Span{x k ∂ x | k ≥ 2} = {v ∈ Vec 1 | Div m (v) = 0}. Now for n ≥ 2. Under the identification Vec 0 n gl(R n ) given by x i ∂ j → e ij , the Euler vector field is the identity 1 and Div m becomes the trace. As ker(f m ) 0 is an ideal in gl(R n ), it can be either 0, R1, sl(R n ) or R1 ⊕ sl(R n ). In the former two cases, Im(f m ) Vec n / ker(f m ), and hence g, would contain sl(R n ) as a subalgebra, contradicting the hypothesis. Hence sl(R n ) ⊆ ker(f m ). If we now show that [Vec n , sl(R n )] = sl(R n ) ∞ k=1 Vec k n , the proof will be complete. Let i = j. We then have [x i ∂ j , x j x α ∂ j ] = (α j + 1)x i x α ∂ j , showing that x i x α ∂ j ∈ [Vec n , sl(R n )]. The only basis elements not of this shape are of the form x k j ∂ j . But define the divergence w.r.t. µ by the requirement that the Lie derivative L v µ equal Div µ (v)µ, then we have σ(v)(fμ) = π * (L v (f µ)) = π * (v(f )µ + Div µ (v)f µ) = v(f )μ + Div µ (v)fμ = ∇ µ v (fμ) + Div µ (v)fμ . This shows that σ(v) = ∇ µ v +ΛDiv µ (v), with Λ = ∂ r , the equivariant vertical vector field defined by the action of R + on ∧ n (T * M ). (Equivariant vertical vector fields on P correspond to sections of ad(P ).) The general case follows by proposition 10.
6.2. Lie algebra cohomology. If we specialise to the case of a trivial bundle over an abelian group G, we find ourselves in the realm of Lie algebra cohomology. The continuous cohomology of the Lie algebra of vector fields with values in the functions has already been unravelled in all degrees [6]. Corollary 5 describes this cohomology only in degree 1, but now with all cocycles rather than just the continuous ones.  Proof. Consider the trivial bundle M × G → M over an abelian Lie group G, which comes equipped with a flat connection ∇ 0 , which acts as Lie derivative. Note that abelian g certainly satisfy the conditions of propositions 10 and 4. View Γ c (ad(P )) C ∞ c (M, g) as a representation of Γ c (T M ), and consider its Lie algebra cohomology. An n-cochain is an alternating linear map Γ c (T M ) n → C ∞ c (M, g). For f 1 ∈ C 1 , closure δf 1 = 0 amounts to Due to this cocycle condition, σ = ∇ 0 + f 1 is once again a Lie algebra homomorphism splitting π * . According to corollary 4, it must therefore take the shape σ = ∇ µ + ΛDiv µ , where Λ ∈ g is constant. One can write ∇ µ = ∇ 0 + ω 1 for some closed 1-form ω 1 , so that f 1 = ω 1 + ΛDiv µ . This classifies the closed 1-cocycles.
Note that a change of density µ = e h µ alters f 1 by a mere coboundary Λdh, so that the choice of µ is immaterial. The class of ω 1 + ΛDiv µ modulo δC 0 is thus determined by [ω 1 ] ∈ H 1 dR (M, g) and Λ ∈ g.
Continuity turns out to be implied by the closedness condition. A similar situation was encountered by Takens in [27], when proving that all derivations of Γ c (T M ) are inner, i.e.