Fiber-Wise Linear Differential Operators

We define a new notion of fiber-wise linear differential operator on the total space of a vector bundle $E$. Our main result is that fiber-wise linear differential operators on $E$ are equivalent to (polynomial) derivations of an appropriate line bundle over $E^\ast$. We believe this might represent a first step towards a definition of multiplicative (resp. infinitesimally multiplicative) differential operators on a Lie groupoid (resp. a Lie algebroid). We also discuss the linearization of a differential operator around a submanifold.


Introduction
Given a vector bundle E → M , it is often interesting to look at geometric structures (functions, vector fields, differential forms, etc.) on the total space E that satisfy appropriate compatibility conditions with the vector bundle structure. Such compatibility is often referred to as linearity in the literature. Accordingly, one speaks about linear functions, linear vector fields, linear differential forms, etc., on the total space of a vector bundle. In the present paper, we will rather use the terminology "fiber-wise linearity", to avoid confusion with other types of linearities. Now, the vector bundle structure is completely determined by (the smooth structure on E) and the action h : R × E → E of the monoid (R, ·) of multiplicative reals by fiber-wise scalar multiplication, h(t, e) = te, for all t ∈ R, and all e ∈ E (see, e.g. [4]). It follows that the fiber-wise linearity of a geometric structure can be usually expressed purely in terms of h. For instance, a function f on E is fiber-wise linear if h * t f = tf for all t. Similarly a vector field X (resp. a differential form ω) on E is fiber-wise linear if h * t X = X for all t = 0 (resp. h * t ω = tω for all t). A fiber-wise linear function is equivalent to a section of the dual vector bundle E * , a fiber-wise linear vector fields is a section of the gauge algebroid of E (see, e.g, [9]) and a fiber-wise linear differential 1-form is equivalent to a section of the first jet bundle J 1 E → M . The latter examples already show that fiber-wise linear structures on E can encode interesting geometric structures on (vector bundles over) M . There are even more interesting examples. A fiber-wise linear symplectic structure ω on E is equivalent to a vector bundle isomorphism E ∼ = T * M . More precisely, there exists a unique fiber-wise linear 1-form ϑ on E such that ω = dϑ, and there exists a unique vector bundle isomorphism E ∼ = T * M that identifies ϑ with the tautological 1-form on T * M , hence ω with the canonical symplectic structure on T * M (see, e.g., [4], see also [11]). There are more examples: a fiber-wise linear metric is equivalent to an isomorphism E ∼ = T * M together with a torsion free connection in T M [12]. As a final remarkable example we recall that a fiber-wise linear Poisson structure on E is the same as a Lie algebroid structure on E * [9].
In this paper we propose the following definition of fiber-wise linear scalar differential operator. An R-linear differential operator ∆ : C ∞ (E) → C ∞ (E) of order q is fiber-wise linear if h * t ∆ = t 1−q ∆ for all t = 0. This definition might seem weird at a first glance. However, it is supported by several different facts. For instance, according to our definition, a function and a vector field are fiber-wise linear if and only if they are fiber-wise linear when regarded as a 0-th order and a first order scalar differential operator, respectively. Moreover the principal symbol of a fiberwise linear differential operator is a fiber-wise linear symmetric multivector. Another supporting remark is that the Laplacian (acting on functions) of a fiber-wise linear metric is a fiber-wise linear differential operator. Finally, a scalar differential operator ∆ can be linearized around a submanifold producing a fiber-wise linear differential operator representing a first order approximation to ∆ in the transverse direction with respect to the submanifold. All these facts suggest that our definition might indeed be the "correct one". Our main result is a description of fiber-wise linear differential operators in terms of somehow simpler data. More precisely we prove the following theorem (see Theorem 6.7 for a more precise statement). This theorem is a little surprising because it describes objects of higher order in derivatives (fiber-wise linear differential operators) in terms of objects of order 1 in derivatives (derivations of an appropriate vector bundle). We hope that this result might be the starting point of a more thorough investigation of multiplicative differential operators on Lie groupoids and, at the infinitesimal level, infinitesimally multiplicative differential operators on Lie algebroids. Multiplicative (resp. infinitesimally multiplicative) structures are geometric structures on a Lie groupoid (resp. Lie algebroid) which are additionally compatible with the groupoid (resp. algebroid) structure. In the last thirty years, starting from the pioneering works of Weinstein on symplectic groupoids [13], multiplicative structures captured the interest of a large community of people working in Poisson geometry and related fields, and today we have a precise description of several different multiplicative structures and their infinitesimal counterparts: infinitesimally multiplicative structures (see [7] for a survey). However, all the examples investigated so far are of order 1 in derivatives and it would be interesting to investigate the compatibility of a Lie groupoid/algebroid with structures of higher order in derivatives, e.g. higher order differential operators. This is a natural issue that might conjecturally lead to new important developments. As infinitesimally multiplicative structures are, in particular, fiber-wise linear structure, this paper might be also considered as a first step in this direction.
The paper is organized as follows. In Section 2 we recall what does it mean for a vector field on the total space of a vector bundle E → M to be fiber-wise linear, i.e. compatible with the vector bundle structure. In Section 3 we discuss fiber-wise linear symmetric multivectors and we describe them in terms of simpler data. This material is well known to experts (although it is scattered in the literature and it is hard to find a universal reference) and the first two sections are mainly intended to fix our notation. In Section 4 we recall what a derivation of a vector bundle E is and introduce what we call E-multivectors, a "derivation analogue" of plain multivectors. We also discuss fiber-wise linear E-multivectors. These objects are not exactly of our primary interest but they play a very useful role in the proofs of our main theorems (Theorems 6.7 and 7.5). To the best of our knowledge the material in the third section is mostly new. Section 5 is an extremely compact introduction to linear differential operators on vector bundles, and, in particular, scalar differential operators. Section 6 contains our main constructions and results: we define and study fiber-wise linear (scalar) differential operators on the total space of a vector bundle E. Somehow surprisingly, fiber-wise linear differential operators on E form a transitive Lie-Rinehart algebra over fiber-wise polynomial functions on E * , with abelian isotropies (Theorem 6.4). The reason is ultimately explained by our main result, Theorem 1.1 above (see also Theorem 6.7 below). As already announced, E-multivectors play a prominent role in the proof. In Section 7 we discuss the linearization of a scalar differential operator ∆ around a submanifold M in a larger manifold. The linearization of ∆ is a fiber-wise linear differential operator on the total space of the normal bundle to M , and can be seen as a first order approximation to ∆ in the direction transverse to M . The existence of a linearization construction strongly supports our definition of fiber-wise linear differential operators.

Core and Linear Vector Fields on a Vector Bundle
As we mentioned in the introduction, the main aim of the paper is to explain what does it mean for a differential operator on the total space E of a vector bundle E → M to be compatible with the vector bundle structure. We will reach our definition (Definition 6.1) by stages. We first need to recall what does it mean for a function, a vector field and, more generally, a multivector on E, to be compatible with the vector bundle structure. We do this in the present and the next section. We adopt the general philosophy of [4,5] where it is shown that a vector bundle structure is encoded in the fiber-wise scalar multiplication, and compatibility with the vector bundle structure is expressed in terms of such multiplication.
So, let π : E → M be a vector bundle. The fiber-wise scalar multiplication by a real number is an action of the multiplicative monoid of reals (R, ·). The algebra C ∞ poly (E) of fiber-wise polynomial functions on the total space E is non-negatively graded: and its k-th homogeneous piece C ∞ (E) k consists of homogeneous polynomial functions of degree k, i.e. functions f ∈ C ∞ (E) such that h * t (f ) = t k f for all t ∈ R. Functions in C ∞ (E) 0 are just (pull-backs via the projection π : E → M of) functions on M . We call them core functions and also denote them by C ∞ core (E). They form a subalgebra in C ∞ poly (E). Functions in C ∞ (E) 1 are fiber-wise linear (FWL for short in what follows) functions, and identify naturally with sections of the dual vector bundle E * . We denote them by C ∞ lin (E). They form a C ∞ core (E) submodule in C ∞ poly (E). We denote by ℓ ϕ the linear function corresponding to the section ϕ ∈ Γ(E * ). The terminology "core function" (and similarly "core vector field", etc., see below) is motivated by the theory of double vector bundles, where "core sections" are sections with an appropriate degree with respect to certain actions of (R, ·) (see, e.g. [9]).
A vector field X ∈ X(E) on E is fiber-wise polynomial, or simply polynomial, if it maps (fiberwise) polynomial functions to polynomial functions. Polynomial vector fields X poly (E) form a (graded) Lie-Rinehart algebra over polynomial functions C ∞ poly (E): We recall for later purposes that a Lie-Rinehart algebra over a commutative algebra A is a vector space L, which is both an A-module and a Lie algebra acting on A by derivations with the following two compatibilities: • the Lie algebra action map ρ : L → Der A is A-linear (it is often called the anchor ), and • the Lie bracket [−, −] : L × L → L is a bi-derivation, i.e. it satisfies the following Leibniz rule: Lie-Rinehart algebras are purely algebraic counterparts of Lie algebroids. For more on Lie-Rinehart algebras, see, e.g. [6] and references therein. Coming back to polynomial vector fields, the k-th homogeneous piece X(E) k of X poly (E) consists of homogeneous "polynomial" vector fields of degree k, i.e. vector fields f ∈ X(E) such that h * t (X) = t k X for all t = 0. Vector fields in X(E) −1 are vertical lifts of sections of E. We call them core vector fields and also denote them by X core (E). They form an abelian Lie-Rinehart subalgebra (beware over the subalgebra C ∞ core (E) = C ∞ (M )) in X poly (E). We denote by e ↑ the vertical lift of a section e ∈ Γ(E).
Vector fields in X(E) 0 are, by definition, fiber-wise linear (FWL) vector fields. They can be equivalently characterized as vector fields preserving linear functions and they satisfy the following property [X, Y ] ∈ X core (E), for all Y ∈ X core (E).
We denote FWL vector fields by X lin (E). They form a Lie-Rinehart subalgebra (over C ∞ core (E)) in X poly (E).
If (x i , u α ) are vector bundle coordinates, then a function f ∈ C ∞ (E) is a core function if and only if f = f (x) and it is a linear functions if and only if, locally, f = f α (x)u α . Similarly, a vector field X ∈ X(E) is a core vector field if and only if, locally, ∂ ∂u α and it is a linear vector field if and only if, locally, Remark 2.1. There is also a useful notion of FWL tensor on E. Let T ∈ Γ(T ⊗r E ⊗ T * ⊗s E) be a tensor field of type (r, s). Then T is a FWL tensor if h * t T = t 1−r T for all t = 0. Notice that FWL tensors are called linear tensor fields in [1], where they are characterized in terms of the fiber-wise addition in E (rather than via the fiber-wise multiplication h as we do). As an instance, consider a metric g ∈ Γ(S 2 T * E). It is linear if h * t g = tg for all t, and it is easy to see that this is in turn equivalent to g being locally of the form Notice that the non-degeneracy condition then implies that the x-dependent matrix (g αi ) is invertible. In particular, the dimension of M and the rank of E must agree. Even more, denoting by T π E = ker dπ the π-vertical bundle, the composition is a vector bundle isomorphism. Here ♭ : T E → T * E is the musical isomorphism, the second and the fourth arrow are those induced by the canonical direct sum decomposition, T E| M = T M ⊕ T π E| M , and the first arrow is the canonical isomorphism. In other words, a non-degenerate symmetric covariant 2-tensor g can only exist on the total space of (a vector bundle isomorphic to) the cotangent bundle. Finally, in standard coordinates (x i , p i ) on T * M , g looks like In particular, g is necessarily of split signature. For more on FWL metrics see [12].

More on FWL Multivector Fields
The material in this section is well-known to experts, and it is partly folklore, partly scattered in the literature. For this reason it is hard to give precise references (the reader may consult, e.g., [8, Appendix A] and references therein, although that reference does not cover the same exact material as the following one). In any case, most of the proofs are straightforward and we omit them.
In particular, l P can be seen as a vector bundle map l P : S k−1 E * → T M , and we will often write l P (ϕ 1 , . . . , ϕ k−1 )(f ) instead of l P (ϕ 1 , . . . , ϕ k−1 , f ). The assignment P → (D P , l P ) establishes a C ∞ (M )-linear bijection between FWL symmetric k-multivectors on E and k-multiderivations of E * , i.e. pairs (D, l) consisting of a map D : Γ(E * ) × · · · × Γ(E * ) → Γ(E * ) and a vector bundle map l : The map l is sometimes called the symbol of D and it is completely determined by D. For this reason, we will often refer to D itself as a k-multiderivation (see, e.g., [3,2] for a skew-symmetric version of multiderivations).
FWL symmetric multivectors on E do also identify with polynomial vector fields on E * . To see this, it is useful to talk about core multivectors first. A k-multivector P on E is core if h * t (P ) = t −k P for all t = 0. Core k-multivectors can be characterized as those multivectors P such that lin (E) and h ∈ C ∞ core (E), and they form a subalgebra X • sym,core (E) in the associative, commutative algebra X • sym (E) (with the symmetric product). More precisely, X • sym,core (E) is the subalgebra spanned by core functions and core vector fields. In particular, X • sym,core (E) identifies with sections Γ(S • E) of the symmetric algebra of E via (e 1 ) ↑ ⊙ · · · ⊙ (e k ) ↑ → e 1 ⊙ · · · ⊙ e k . e i ∈ Γ(E). In its turn, Γ(S • E) identifies with polynomial functions on E * in the via the (degree preserving) algebra isomorphism and, in what follows, we will often understand the latter identifications. Notice that the resulting isomorphism We can now go back to FWL symmetric multivectors. The symmetric product of a core symmetric multivector and a FWL one is a FWL multivector, and this turns It is easy to see that the Poisson bracket H P = {P, −} preserves core multivectors. Hence, it is a derivation of the commutative algebra X • sym,core (E) ∼ = C ∞ poly (E * ). In its turn, H P extends uniquely to a polynomial vector field, also denoted H P , on E * . The assignment P → H P establishes a degree inverting isomorphism of Lie algebras, between the Lie algebra of linear symmetric multivectors on E (with the Poisson bracket) and polynomial vector fields on E * (with the commutator). When we equip X • sym,lin (E) with the symmetric product by a core multivector, the latter isomorphism becomes an isomorphism of Lie-Rinehart algebras.
Finally, we remark that linear symmetric multivectors fit in the following short exact sequence where the second arrow identifies the section

More on Derivations of a Vector Bundle
In this section, for a vector bundle V → M , we introduce a notion of (symmetric) V -multivector (Definition 4.1). To the best of our knowledge this notion is new. It will play a significant role in the description of fiber-wise linear differential operators provided in Section 6. Symmetric Vmultivectors are in many respect similar to plain symmetric multivectors, so the proofs of most of the statements in this section parallel the proofs of the analogous statements for multivectors and we omit them.
We begin with a vector bundle E → M and remark that the space X 1 lin (E) = X lin (E) of linear vector fields is of particular interest. The assignment X → D X establishes an isomorphism of Lie-Rinehart algebras (over C ∞ (M )) between linear vector fields on E and derivations of E * , i.e. 1-multiderivations. We stress that In the following, we denote by D(V ) the Lie-Rinehart algebra of derivations of a vector bundle V . It is the Lie-Rinehart algebra of sections of a Lie algebroid DV → M whose Lie bracket is the commutator of derivations and whose anchor is the symbol map D → l D .
Notice also that the assignment X → H X (see the last paragraph of the previous section) does also establish an isomorphism of Lie-Rinehart algebras (over C ∞ (M )) between linear vector fields on E and linear vector fields on E * . Accordingly we have a canonical Lie algebroid isomorphism DE → DE * , D → D * which is explicitly given by for every ϕ ∈ Γ(E * ), e ∈ Γ(E), where −, − : E * ⊗ E → R M := M × R is the duality pairing. In the following we will simply denote by D the derivation of E * induced by a derivation of E (and vice-versa). It is easy to see that More generally, a derivation D of a vector bundle V induces a derivation, also denoted D, in each component of the whole (symmetric, resp. alternating) tensor algebra of V ⊕ V * . The latter derivation is defined imposing the obvious Leibniz rule with respect to the tensor product and the contraction by an element in the dual.
We are now ready to define the algebra of V -multivectors, which is a "derivation analogue" of the Poisson algebra of symmetric multivectors. So, let V → M be a vector bundle, and consider the graded spaceD where the tensor product is over functions on M . Consider the graded subspace D • sym (V ) ⊂D • consisting of elements projecting on symmetric multivectors fits in an exact sequence: A symmetric k-V -multivector D ∈ D k sym (V ) will be often interpreted as an operator (1) the associative product given by Here Proof. A long and tedious computation that we omit.
as an abelian subalgebra and an ideal.
There is also a notion of FWL V -multivector. In order to discuss it, it is is useful to discuss derivations of pull-back vector bundles first. So, let V be a vector bundle, and consider its pull-back V P := π * V along a surjective submersion π : P → M . Clearly, a derivation D of V P is completely determined by its symbol and its action on pull-back sections. The restriction D M := D| Γ(V ) of D to pull-back sections is a derivation along π, i.e. it is an R-linear map D M : Γ(V ) → Γ(V P ) and there exists a, necessarily unique, vector field along π, denoted l D M ∈ Γ(π * T M ), fitting in the Leibniz rule The correspondence D → (l D , D M ) establishes a C ∞ (M )-linear bijection between derivations D of V P and pairs (X, D M ) consisting of a vector field X ∈ X(P) and a derivation along π satisfying the following additional compatibility: where the tensor product is over C ∞ (M ), and multiplicative reals act on Γ(V E ) via their action on the first factor. As for functions, we denote by h * this action.
and they correspond to pairs (X, D M ) where X ∈ X poly (E) and D M takes values in polynomial sections.
We can also consider polynomial symmetric V E -multivectors. We will only need core and FWL ones.
) the space of FWL (resp. core) symmetric V E -multivectors. The Lie bracket {−, −} on V -multivectors preserves FWL ones. Additionally, the projection L : and we get a short exact sequence of Lie algebras: Proof. Straightforward.
It easily follows from the above proposition that a core k-V E -multivector is completely determined by its symbol. More precisely, the symbol map D → L D establishes a one-to-one correspondence between core k-V E -multivectors and core multivectors. We conclude that D Proof. Straightforward.
In particular, a FWL symmetric k-V -multivector D determines a map: for all ϕ i ∈ Γ(E * ), and all w ∈ Γ(V ). The map Φ D is C ∞ (M )-multilinear and symmetric. Hence, it can be seen as a vector bundle map Φ D : S k−1 E * → DV , or, equivalently, as a section of Proof. Easy and left to the reader.
According to Proposition 4.6 we will sometimes call the pair (L D , Φ D ) itself a FWL symmetric V E -multivector. Now, we can combine the exact sequences (3.2) and (4.3) in one exact commutative diagram: We only need to explain the map I. To do that, we first remark that π-vertical vector fields act naturally on sections of V E , via and v ∈ Γ(V ). Now, take e 1 , . . . , e k ∈ Γ(E) and ϕ ∈ Γ(E * ). Then Equivalently, we can interpret e 1 ⊙· · ·⊙e k as a core V E -multivector, via the isomorphism D • sym,core (V E ) ∼ = Γ(S • E), and then multiply by the FWL function ℓ ϕ to get a FWL V E -multivector.
We conclude this section showing that FWL symmetric V E -multivectors do also identify with polynomial derivations of V E * . This is an easy consequence (among other things) of Proposition 4.6. Indeed, take D ∈ D • sym,lin (V E ) and let (L D , Φ D ) be the corresponding pair. Denote by π : E * → M the projection. We claim that Φ D can be seen as a derivation along π. Indeed Φ D is a section of S •−1 E ⊗ DL and, by acting on a section v ∈ Γ(V ) with the DL-factor, we get a section Φ D (v) of S •−1 E ⊗ V , i.e. a polynomial section of Γ(V E * ). In the following, we use this construction to interpret Φ D as a derivation along π. If we do so, the pair (H L D , Φ D ) consists of a vector field on E * , and a derivation Φ D along π, with the additional property that dπ • H L D = l • Φ D , hence it corresponds to a (polynomial) derivation D * of the pull-back vector bundle V E * . Finally a tedious, but straightforward computation shows that the bijection D → D * between linear V E -multivectors and polynomial derivations of V E * obtained in this way do also preserve the Lie algebra structures. When we equip D • sym,lin (E) with the product by a core V E -multivector, the latter bijection becomes an isomorphism of Lie-Rinehart algebras. We have thus proved the main result in this section:

Differential Operators and Their Symbols
We finally come to the object of our primary interest: differential operators. This sections is a super-short review of the subject.
Let V, W → M be vector bundles. A (linear) differential operator (DO in the following) of order q from V to W is an R-linear map ∆ : In particular, DO of order zero are just vector bundle maps V → W . We denote by DO q (V, W ) the space of order q DOs from V to W . Clearly, a DO of order q is also a DO of order q + 1, and we get the filtration The union of all DO q (V, W ) will be denoted simply by DO(V, W ). A scalar DO on M is a DO acting on functions over M , i.e. a DO from the trivial line bundle R M := M × R to itself. We use the symbol DO q (R E ) (instead of DO q (R M , R M )) for scalar DOs.
The composition of an order q and an order r DO is an order q + r DO. In particular, for all q, DO q (V, W ) is a C ∞ (M )-module in two different ways: via composition on the left and composition on the right with a function on M (seen as an order 0 DO). We will consider the first module structure unless otherwise stated. The space DO(R M ) is a filtered non-commutative algebra with the composition. It is actually the universal enveloping algebra of the tangent Lie algebroid T M → M . Being an associative algebra, DO(R M ) is also a Lie algebra with the commutator. Notice that the commutator of an order q and an order r scalar DO is an order q + r − 1 scalar DO.
Given an order q DO ∆ : Γ(V ) → Γ(W ) from V to W , and functions f 1 , . . . , f q , the nested commutator is an order 0 DO. Additionally, it is a derivation in each of the arguments f i and it is symmetric in those argument. In this way, we get a map The map σ is called the symbol and it fits in a short exact sequence of C ∞ (M )-modules where the second arrow is the inclusion. We conclude this short review section commenting briefly on the coordinate description of (scalar) DOs. To do this we first fix our conventions on the multi-index notation for multiple partial derivatives. Let (x i ), i = 1, . . . , n be variables. A length k multi-index I is a word I = i 1 . . . i k , with i j = 1, . . . , n, where words are considered modulo permutations of their letters. The length k of a multi-index I = i 1 · · · i k is also denoted |I|. Words can be composed by concatenation and we also consider the empty multi-index ∅. If we do so, then multi-indexes are elements in the free abelian monoid spanned by 1, . . . , n. The lenght is then a monoid homomorphism. A lenght k multi-index I = i 1 . . . i k , determines an order k DO where, for a multi-index I, we denoted by I! the product I[1]! · · · I[n]! where I[i] is the number of times the letter i occurs in I. Finally, if ∆ is an order q scalar DO locally given by (5.2), then its symbol σ(∆) is locally given by

Core and Fiber-wise Linear Differential Operators
This is the main section of the paper. We propose a notion of FWL (scalar) DO on the total space of a vector bundle. Our definition is partly motivated by the fact that the symbol of a FWL DO is a FWL multivector. It is also motivated by the linearization construction discussed in the next section. Yet another motivating little fact is that the Laplacian of a FWL metric is a FWL DO (Example 6.2).
Let E → M be a vector bundle. We have learnt from Sections 2, 3 and 4 that, given a type T of geometric structures on manifolds (functions, vector fields, tensors, etc.) appropriate notions of core and FWL structures of the type T on E exist, and these notions can be identified by means of the following recipe: 1) notice that the space T(E) of structures of type T on E is naturally graded (via the action of multiplicative reals on E by fiber-wise scalar multiplication), 2) identify the smallest degree k for which the degree k homogeneous component T(E) k of T(E) is non-trivial, and 3) put T core (E) = T(E) k and T lin (E) = T(E) k+1 . A quick check shows that this recipe cooks up the required definitions in all the cases considered so far. Notice that we could make this recipe much more rigorous adopting for the rather vague "geometric structure of type T" the very precise notion of natural vector bundle T, but we will not need this level of abstraction. We adopt the strategy described above to define core and FWL DOs on E. Consider the noncommutative algebra DO(R E ) of scalar DOs ∆ : C ∞ (E) → C ∞ (E). We begin noticing that, for each q, the space DO q (R E ) of DOs ∆ : C ∞ (E) → C ∞ (E) of order q is naturally graded: where DO q (R E ) k consists of degree k DOs (of order q), i.e. DOs ∆ such that h * t (∆) = t k ∆ for all t = 0. The smallest degree k for which DO q (R E ) k is non-trivial is k = −q. So, following our recipe, we put DO q,core (E) := DO q (R E ) −q , and call them core DOs. We also put Let (x i , u α ) be vector bundle coordinates on E, and let (x i , u α ) be dual coordinates on E * . A DO F ∈ DO q (R E ) is a core DO if and only if, locally, where A = α 1 · · · α q is a lenght q multi-index. It follows from (6.2) that DO core (E) ⊂ DO(R E ) is the subalgebra spanned by core functions C ∞ core (M ) and core vector fields X core (E). Equivalently, it is the universal enveloping algebra of the abelian Lie algebroid E ⇒ M . Because of the latter description, there is an algebra isomorphism Γ(S • E) → DO core (E), mapping a monomial e 1 ⊙ · · · ⊙ e q , e i ∈ Γ(E), to the DO e ↑ 1 • · · · • e ↑ q . In its turn, as already mentioned, Γ(S • E) identifies with polynomial functions on E * . In the following we will often identify DO core (E) with both Γ(S • E) and C ∞ poly (E * ) via the latter isomorphisms. If F ∈ DO q,core (E) is locally given by (6.1), then it identifies with where, for A = α 1 · · · α q , we denoted by u A the monomial u α 1 · · · u αq . We will always consider DO(R E ) as a DO core (E)-module with the scalar multiplication given by the left composition.
We now pass to FWL DOs. Following our recipe again, for each q we put DO q,lin (E) := DO q (R E ) −q+1 .
Definition 6.1. DOs in DO q,lin (E) are called fiber-wise linear differential operators (FWL DOs) of order q.
For instance, DO 0,lin (E) = C ∞ lin (E), and DO 1,lin (E) = X lin (E) ⊕ C ∞ (M ). It is also clear that DO q,lin (E) ⊃ DO q−1,core for all q. More precisely is FWL if and only if, in vector bundle coordinates, it looks like It is easy to see from this formula that DO lin (E) ⊂ DO(R E ) is the DO core (E)-submodule spanned by 1, C ∞ lin (E) and X lin (E). Example 6.2. Let g be a metric on E, and assume it is FWL. Then, the associated Laplacian operator ∆ g : C ∞ (E) → C ∞ (E) is a FWL DO operator (of order 2). One can see this working in vector bundle coordinates. But there is also a (basically) coordinate free proof that we now illustrate. First of all, from g being FWL, it immediately follows that the inverse tensor g −1 is FWL as well. Now, the covariant derivative ∇ϑ of a 1-form ϑ along the Levi-Civita connection ∇ is the covariant 2-tensor given by the formula: where ♯ : T * E → T E is the musical isomorphism. Equivalently, the covariant derivative ∇ X Y of a vector field Y along another vector field X is the vector field ∇ X Y that acts on functions f ∈ C ∞ (M ) as follows: where grad f = ♯(df ) is the gradient of f . Using (6.4) (or (6.5)) and the naturality of both the de Rham differential and the Lie derivative, it is easy to see that the covariant derivative of arbitrary tensor fields commutes with the pull back along h t for all t = 0. As the Laplacian ∆ g f of a function f is obtained by contracting the covariant derivative of df with g −1 , then ∆ g decreases by one the degree of a homogeneous (fiber-wise polynomial) function. So it is a second order DO of degree 1 − 2 = −1, i.e. a FWL DO of order 2, as claimed. It might be also interesting to remark that the Levi-Civita connection of a FWL metric is a FWL connection according to a definition introduced in [10]. Proof. The "only if part" of the statement immediately follows from an obvious order/degree argument. For the "if part", consider a DO ∆ of order r. Locally, Assume that ∆ is in the stabilizer of DO core (E). We want to show that ∆ is the sum of operators of the form (6.3) (with possibly varying q ≤ r). As x i is a core function for all i, the commutator [∆, x i ] is a core DO. But so it can only be a core DO for all i if 1) ∆ I|A (x, u) = 0 for |I| > 1, and 2) ∆ i|A (x, u) = ∆ i|A (x). In other words, ∆ is necessarily of the form and, to conclude, it is enough to prove that ∆ . To do this, recall that ∂ ∂u α is a core vector field for all α, hence ∆, ∂ ∂u α is a core DO. But, from (6.6), ) is a (non-necessarily homogeneous) first order polynomial in the variables u, as desired.
It follows from Lemma 6.3 that DO lin (E) is a Lie subalgebra in DO(R E ). As already mentioned, it is also a DO core -submodule. Actually, it is a Lie-Rinehart algebra over DO core (E), the anchor being the adjoint operator ad : ∆ → ad(∆) := [∆, −]. To see this, first notice that ad(∆) is indeed a well-defined derivation of DO core (E) for all ∆ ∈ DO lin (E). Now, take ∆, ∆ ′ ∈ DO lin (E) and F, F ′ ∈ DO core (E), and compute Finally, from DO core (E) ∼ = C ∞ poly (E * ), we see that, for every ∆ ∈ DO lin (E), the derivation ad(∆) determines a polynomial vector field (of the same degree) on E * , also denoted ad(∆).
Theorem 6.4. The sequence of Lie-Rinehart algebras Proof. First of all, as already remarked, DO core (E) is in DO lin (E). Even more, as it is an abelian subalgebra in DO(R E ), then it is actually in the kernel of ad : DO lin (E) → X poly (E * ). To see that core DOs exhaust the kernel of ad (i.e. DO core (E) is its own centralizer), assume that [∆, F ] = 0 for all F ∈ DO core (E). Then, exactly the same computation as in the proof of Lemma 6.3 shows that ∆ is locally of the form (6.3) with ∆ i|A (x) = ∆ B α (x) = 0, i.e. ∆ ∈ DO core (E). For the exactness of the sequence (6.7) it remains to show that the map ad : DO lin (E) → X poly (E * ) is surjective. To do that, we work in local coordinates again. So, let (x i , u α ) be vector bundle coordinates on E, and let (x i , u α ) be dual coordinates on E * . It is not hard to see that, if ∆ is locally given by (6.3), then the vector field ad(∆) is locally given by (6.8) ad where, for a multi-index A = α 1 · · · α s , we denoted by u A the monomial u α 1 · · · u αs (s = q, q − 1). As (6.8) is the local expression of a generic homogeneous polynomial vector field of degree q − 1, we are done.
Our next aim is proving that the Lie-Rinehart algebra DO lin (E) is canonically isomorphic to the Lie-Rinehart algebra of polynomial derivations of an appropriate line bundle on E * . We begin with a simple Proposition 6.5. The symbol σ(∆) of a FWL DO ∆ ∈ DO q,lin (E) is a FWL symmetric qmultivector field. Every FWL symmetric q-multivector field is the symbol of an order q FWL DO.
Proof. The statement immediately follows from (6.3) and the easy fact that a symmetric q-multivector P is FWL if and only if, in vector bundle coordinates, it is of the form Now let ∆ ∈ DO q,lin (E). Notice that the adjoint operator ad(∆), seen as a polynomial vector field on E * , corresponds exactly to the symbol σ(∆) via the isomorphism X q sym,lin (E) ∼ = X(E * ) q−1 . It is also clear that, in view of its coordinate form (6.3), ∆ is completely determined by σ(∆) or, equivalently, ad(∆), together with the map The map Ψ ∆ is clearly well-defined. Additionally, it enjoys the following properties (1) Ψ ∆ is symmetric, (2) Ψ ∆ is a first order DO in each entry. More precisely, we have the following Lemma 6.6. The map Ψ ∆ satisfies Proof. Let ϕ i and f be as in the statement, and compute It remains to compute the last summand. So where we used that, from (6.3) again, the last summand in the second line is necessarily zero. This concludes the proof.
Only the terms with l = p, p − 1 (hence m = q − 1, q, respectively) survive, and we get We already computed the second summand, while the first summand is From (6.13) and (6.14) it easily follows that Φ F •∆ = F · Φ ∆ as claimed.
The surjectivity of the map ∆ → (σ(∆), Φ ∆ ) now follows from (local) dimension counting. It remains to check that the isomorhism A : DO lin (E) → D sym,lin (L E ) defined in this way is both anchor and bracket preserving. For the anchor, the anchor of (σ(∆), Φ ∆ ) is the derivation of DO core (E) = Γ(S • E) = C ∞ poly (E) corresponding to the linear multivector σ(∆), which is exactly ad(∆).
For the bracket, as we already discussed C ∞ poly (E)-linearity and compatibility with the anchor, it is enough to discuss the brackets of generators. As already remarked, DO lin (E) is generated (over DO core (E)) by 1, C ∞ lin (E) and X lin (E). A direct check shows that for all ϕ ∈ Γ(E * ) and all X ∈ X lin (E). Here X * and D X are, respectively, the linear vector field on E * , and the derivation of L = ∧ top E corresponding to X. It is now easy to check that the brackets are preserved on these generators, and this concludes the proof.
Composing with the isomorphism D • sym,lin (L E ) ∼ = D(L E * ) we get a (degree inverting) Lie-Rinehart algebra isomorphism that we denote by A again.
Remark 6.8. Let (x i , u α ) be vector bundle coordinates on E, and let (x i , u α ) be dual coordinates on E * . Denote by Vol u = u 1 ∧ · · · ∧ u top the local coordinate generator of Γ(L). It is easy to check using, e.g., (6.8), (6.15), and the C ∞ poly (E * )-linearity, that, if the operator ∆ ∈ DO lin (E) is locally given by (6.3), then the corresponding derivation A(∆) ∈ D(L E * ) maps a local section

Linearization of Differential Operators
Let E be a manifold, let M ⊆ E be a submanifold and let ∆ ∈ DO(R E ) be a scalar DO. Denote by E → M the normal bundle to M , i.e. E = T E| M /T M . In this section we show that, under appropriate linearizability conditions, the DO ∆ can be linearized around M yielding a FWL differential operator ∆ lin ∈ DO lin (E). The DO ∆ lin represents the first order approximation of ∆ around M in the direction transverse to M . This linearization construction is a further motivation supporting our definition of FWL DOs.
So let M ⊆ E be a submanifold and let E → M be its normal bundle. We will often consider adapted coordinates on E around points of M , i.e. coordinates (X i , U α ) such that M : {U α = 0 . In particular, the restrictions (x i = X i | M ) are coordinates on M . From U α | M = 0 we see that u α := dU α | M are conormal 1-forms and (x i , u α ) are vector bundle coordinates on E.
We want to explain what does it mean to linearize an order q DO operator ∆ ∈ DO q (R E ) around M . We proceed as follows: 1) first, we recall the linearization of a function, 2) second, we discuss the linearization of a symmetric multivector, and, finally 3) we define the linearization of a generic DO. So, let F ∈ C ∞ (E). We say that F is linearizable (around M ) if F | M = 0. In this case, dF | M is a conormal 1-form to M , i.e. a section of the conormal bundle E * ֒→ T * E| M . Hence it corresponds to a FWL function on E. We put F lin = ℓ dF | M and call it the linearization of F . For instance, if (X i , U α ) are adapted coordinates on E, then the (U α ) are linearizable and the linear fiber coordinates (u α ) on E are their linearizations. If F is any linearizable function on E, then locally, around a point of M , F (X, U ) = F α (X)U α + O(U 2 ), for some functions F α (X) of the (X i ) (given by F α (X) = ∂F ∂U α (X, 0)), and, in this case, F lin = F α (x)u α . Notice that every linear function ϕ ∈ C ∞ lin (E) is the linearization of a (non-unique) linearizable function F ∈ C ∞ (E): ϕ = F lin . We now pass to symmetric multivectors. So, let P be a symmetric q-multivector on E. We say that P is linearizable if it belongs to the ideal I M in X • sym (E) spanned by vector fields that are tangent to M . In other words M is a coisotropic submanifold of E with respect to P . This notion of linearizable multivector agrees with that of FWLizable tensor described in [10,Definition 5.4].
Proposition 7.1. Let P ∈ X q sym (E) be a linearizable symmetric multivector on E. Then, there exists a unique FWL symmetric q-multivector P lin on E such that for all linearizable functions F i ∈ C ∞ (E). The linearization P → P lin preserves the Poisson bracket of symmetric multivectors.
Proof. We begin remarking that, as P ∈ I M , the function P (F 1 , . . . , F q ) is clearly linearizable for any choice of linearizable functions F i . Now we want to show that the rhs of (7.1) does only depend on (F i ) lin . We do this in coordinates. So, let (X i , U α ) be adapted coordinates on E, and let (x i , u α ) be the associated vector bundle coordinates on E. Locally It follows from P ∈ I M that P α 1 ,...,αq (X, 0) = 0. Now compute Using that F i | M = 0, and that P α 1 ,...,αq (X, 0) = 0, we find which does only depend on the (F i ) lin . As every FWL function is a linearization, this also shows that P lin is well-defined on linear functions. Finally, P lin can be uniquely extended to all functions on E, as a symmetric q-multivector, also denoted by P lin , and locally given by In particular P lin is a linear multivector.
For the last part of the statement, first notice that the ideal I M is preserved by the Poisson bracket, so, if P, Q ∈ I M , then it makes sense to linearize {P, Q}. The rest follows easily from Equation (7.1).
Remark 7.2. Proposition 7.1 is a "symmetric multivector analogue" of the following well-known fact. Let (P, π) be a Poisson manifold, let (T * P) π be its cotangent algebroid and let M ⊆ P be a coisotropic submanifold. Then the conormal bundle N * M ⊆ T * P is a subalgebroid (T * P) π , hence the normal bundle N M is equipped with a linear Poisson structure π lin (see, e.g., [14,Section 3]).
By definition, the multivector P lin in Proposition 7.1 is the linearization of P . We finally come to a generic differential operator ∆ ∈ DO q (R E ).
• An order q − 1 DO ∆ ∈ DO(R E ) is always order q linearizable, but it might not be order q − 1 linearizable.
is order 1 linearizable if and only if it is tangent to M , i.e. it is a linearizable vector field. Now, let ∆ ∈ DO q (R E ) be an order q linearizable DO. In order to linearize it, we use Theorem 6.7. In other words, from ∆ we cook up a q-L E -multivector (P lin , Φ) of L E * = E * × M L, with L = ∧ top E as in Section 6. The construction is inspired by the proof of Theorem 6.7 itself. We let P lin ∈ X q sym,lin (E) be the linearization of the symbol P = σ(∆) of ∆. It remains to define the vector bundle map Φ : S q−1 E * → DL.
To compute T q , we use a simple trick: for every two scalar DOs , ′ we have