We define a new notion of fiberwise linear differential operator on the total space of a vector bundle E. Our main result is that fiberwise linear differential operators on E are equivalent to (polynomial) derivations of an appropriate line bundle over . 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.
Given a vector bundle , 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 “fiberwise 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 of the monoid of multiplicative reals by fiberwise scalar multiplication: for all and all (see, e.g., ). It follows that the fiberwise linearity of a geometric structure can usually be expressed purely in terms of h. For instance, a function f on E is fiberwise linear if and only if for all t. Similarly, a vector field X (resp. a differential form ω) on E is fiberwise linear if and only if for all (resp. for all t). A fiberwise linear function is equivalent to a section of the dual vector bundle , a fiberwise linear vector field is equivalent to a section of the gauge algebroid of E (see, e.g., ), and a fiberwise linear differential 1-form is equivalent to a section of the first jet bundle . The latter examples already show that fiberwise linear structures on E can encode interesting geometric structures on (vector bundles over) M. There are even more interesting examples. A fiberwise linear symplectic structure ω on E is equivalent to a vector bundle isomorphism . To see this, notice that ω, together with the standard direct sum decomposition of the tangent bundle of E restricted to the zero section, gives rise to a bilinear map . As ω is non-degenerate, the induced map is a vector bundle isomorphism. From closedness and fiberwise linearity, , the pull-back along of the canonical 2-form on . It is clear that is the unique vector bundle isomorphism with this property (see, e.g., , see also ).
Another interesting example is the following: a fiberwise linear metric is equivalent to an isomorphism together with a torsion free connection in TM .
As a final remarkable example, we recall that a fiberwise linear Poisson structure on E is the same as a Lie algebroid structure on .
In this paper, we propose the following definition of fiberwise linear scalar differential operator. An -linear differential operator of order q is fiberwise linear if for all . 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 fiberwise linear if and only if they are fiberwise 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 fiberwise linear symmetric multivector field. Another supporting remark is that the Laplacian (acting on functions) of a fiberwise linear metric is a fiberwise linear differential operator. Finally, a scalar differential operator Δ can be linearized around a submanifold producing a fiberwise linear differential operator representing a first order approximation to Δ 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 fiberwise 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).
Let be a vector bundle. Then there is a degree inverting -linear bijection between fiberwise linear scalar differential operators and polynomial derivations of the line bundle .
This theorem is a little surprising because it describes objects of higher order in derivatives (fiberwise 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 , 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  for a survey). However, all 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, fiberwise linear structures, 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 it means for a vector field on the total space of a vector bundle to be fiberwise linear, i.e. compatible with the vector bundle structure. In Section 3, we discuss fiberwise linear symmetric multivector fields 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 Sections 2 and 3 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 multivector fields. We also discuss fiberwise 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 Section 4 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 fiberwise linear (scalar) differential operators on the total space of a vector bundle E. Somehow surprisingly, fiberwise linear differential operators on E form a transitive Lie–Rinehart algebra over fiberwise polynomial functions on , with abelian isotropies (Theorem 6.4). The reason is ultimately explained by our main result, the theorem 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 Δ around a submanifold M in a larger manifold. The linearization of Δ is a fiberwise linear differential operator on the total space of the normal bundle to M, and can be seen as a first order approximation to Δ in the direction transverse to M. The existence of a linearization construction strongly supports our definition of fiberwise linear differential operators.
We will adopt systematically the Einstein convention on the sum of repeated upper-lower indexes. We will not adopt the Einstein convention for sums over (repeated) multi-indexes.
2 Core and linear vector fields on a vector bundle
As we mentioned in Section 1, the main aim of this paper is to explain what it means for a differential operator on the total space E of a vector bundle to be compatible with the vector bundle structure. We will reach our definition (Definition 6.1) by stages. We first need to recall what it means for a function, a vector field and, more generally, a multivector field 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 fiberwise scalar multiplication, and compatibility with the vector bundle structure is expressed in terms of such multiplication.
So, let be a vector bundle. The fiberwise scalar multiplication by a real number
is an action of the multiplicative monoid of reals . The algebra of fiberwise polynomial functions on the total space E is non-negatively graded:
and its k-th homogeneous piece consists of homogeneous polynomial functions of degree k, i.e. functions such that
for all . Functions in are just (pull-backs via the projection of) functions on M. We call them core functions and also denote them by . They form a subalgebra in . Functions in are fiberwise linear (FWL for short in what follows) functions, and identify naturally with sections of the dual vector bundle . We denote them by . They form a -submodule in . We denote by the linear function corresponding to the section . 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 (see, e.g., ).
A vector field on E is fiberwise polynomial, or simply polynomial, if it maps (fiberwise) polynomial functions to polynomial functions. Polynomial vector fields form a (graded) Lie–Rinehart algebra over polynomial functions :
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 is A-linear (it is often called the anchor).
The Lie bracket 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.,  and the references therein.
Coming back to polynomial vector fields, the k-th homogeneous piece of consists of homogeneous polynomial vector fields of degree k, i.e. vector fields such that
for all . Vector fields in are vertical lifts of sections of E. We call them core vector fields and also denote them by . They form an abelian Lie–Rinehart subalgebra (note: over the subalgebra ) in . We denote by
the vertical lift of a section .
Vector fields in are, by definition, fiberwise linear (FWL) vector fields. They can be equivalently characterized as vector fields preserving linear functions and they satisfy the following property:
We denote FWL vector fields by . They form a Lie–Rinehart subalgebra (over ) in .
If are vector bundle coordinates, then a function is a core function if and only if, locally, and it is a linear function if and only if, locally, . Similarly, a vector field is a core vector field if and only if, locally,
and it is a linear vector field if and only if, locally,
There is also a useful notion of FWL tensor on E. Let be a tensor field of type .
Then is a FWL tensor if
for all .
Notice that FWL tensor fields are called linear tensor fields in , where they are characterized in terms of the fiberwise addition in E (rather than via the fiberwise multiplication h as we do).
As an instance, consider a metric . It is linear if 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 is invertible. In particular, the dimension of M and the rank of E must agree. Even more, by denoting by the π-vertical bundle, the composition
is a vector bundle isomorphism. Here is the musical isomorphism, the second and the fourth arrow are those induced by the canonical direct sum decomposition, , 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 on , g looks like
for some appropriate local functions . In particular, g is necessarily of split signature. For more on FWL metrics, see .
3 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 the references therein, although that reference does not cover the same exact material as the present one). In any case, most of the proofs are straightforward and we omit them.
We will need to consider FWL symmetric multivector fields. According to Definition 2.1, a k-multivector field P on the total space E of a vector bundle is FWL if
for all . We denote by FWL symmetric multivector fields.
There is a useful characterization of FWL k-multivector fields. Namely, a k-multivector field P on E is FWL if and only if
for all and all . In particular, an FWL symmetric k-multivector field P determines a pair of maps :
for all and all . The maps satisfy the following properties:
is -multilinear and symmetric.
is -multilinear and symmetric in the first -arguments.
is a derivation in its last argument.
In particular, can be seen as a vector bundle map , and we will often write
instead of . The assignment establishes a -linear bijection between FWL symmetric k-multivector fields on E and k-multiderivations of , i.e. pairs consisting of a map
and a vector bundle map (equivalently, a section of ) satisfying
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).
Recall that there is a natural Poisson bracket on symmetric multivector fields given by the following (Gerstenhaber-type) formula:
for all -multivector fields , all -multivector fields , and all functions , where denotes -unshuffles. The Poisson bracket (3.1) preserves FWL symmetric multivector fields, and the Poisson bracket of FWL Poisson multivector fields identifies with the obvious Gerstenhaber-like bracket of the associated multiderivations .
FWL symmetric multivector fields on E do also identify with polynomial vector fields on . To see this, it is useful to talk about core multivector fields first. A k-multivector field P on E is core if
for all . Core k-multivector fields can be characterized as those multivector fields P such that
for all and , and they form a subalgebra in the associative, commutative algebra (with the symmetric product). More precisely, is the subalgebra spanned by core functions and core vector fields. In particular, identifies with sections of the symmetric algebra of E via
with (recall that denotes the vertical lift of a section ). In its turn, identifies with polynomial functions on via the (degree preserving) algebra isomorphism
and, in what follows, we will often understand the latter identifications. Notice that the resulting isomorphism
is given by
where , and .
We can now go back to FWL symmetric multivector fields. The symmetric product of a core symmetric multivector field and an FWL one is an FWL multivector field, and this turns into a -module. Now, let . It is easy to see that the Poisson bracket preserves core multivector fields. Hence, it is a derivation of the commutative algebra . In its turn, extends uniquely to a polynomial vector field, also denoted by , on . The assignment
establishes a degree inverting isomorphism of Lie algebras, between the Lie algebra of linear symmetric multivector fields on E (with the Poisson bracket) and polynomial vector fields on (with the commutator). When we equip with the symmetric product by a core multivector field, the latter isomorphism becomes an isomorphism of Lie–Rinehart algebras. We summarize and complement the above discussion on core and FWL multivector fields with the following proposition.
Let be a vector bundle. The map
given by (3.2) is a degree inverting isomorphism of (graded) commutative algebras, and the map
is a degree inverting isomorphism of (graded) Lie–Rinehart algebras over .
Let be vector bundle coordinates on E and let be the corresponding dual vector bundle coordinates on . Then every core symmetric q-multivector field is locally of the form
and we have
Additionally, every FWL symmetric q-multivector field is locally of the form
and we have
The proof of those parts of the statement that have not been already discussed is straightforward, and we leave the easy details to the reader. ∎
Finally, we remark that linear symmetric multivector fields fit in the following short exact sequence:
where the second arrow identifies the section of with the FWL k-multivector field .
4 More on derivations of a vector bundle
In this section, for a vector bundle , 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 fiberwise linear differential operators provided in Section 6. Symmetric V-multivectors are in many respects similar to plain symmetric multivector fields, so the proofs of most of the statements in this section parallel the proofs of the analogous statements for multivector fields, and we usually omit them.
We begin with a vector bundle and remark that the space of linear vector fields is of particular interest. The assignment establishes an isomorphism of Lie–Rinehart algebras (over ) between linear vector fields on E and derivations of , i.e. 1-multiderivations. We stress that
In the following, we denote by the Lie–Rinehart algebra of derivations of a vector bundle V. It is the Lie–Rinehart algebra of sections of a Lie algebroid whose Lie bracket is the commutator of derivations and whose anchor is the symbol map .
Notice also that the assignment (see Proposition 3.1) does also establish an isomorphism of Lie–Rinehart algebras (over ) between linear vector fields on E and linear vector fields on . Accordingly, we have a canonical Lie algebroid isomorphism
which is explicitly given by
for every and , where is the duality pairing. In the following, we will simply denote by D the derivation of 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 by D, in each component of the whole (symmetric, resp. alternating) tensor algebra of . 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 multivector fields. So, let be a vector bundle, and consider the graded space , where the tensor product is over functions on M. Notice that is concentrated in positive degrees. Consider the graded subspace consisting of elements projecting on symmetric multivector fields via
Finally, we define the graded space by putting
We denote by
the projection (given simply by the identity in degree 0). Clearly, fits in an exact sequence:
Elements in are symmetric k-V-multivectors.
For , a symmetric k-V-multivector will often be interpreted as an operator
As such it is symmetric and a derivation in the first arguments. Additionally, there exists a (necessarily unique) symmetric multivector field such that
for all and .
The space of symmetric V-multivectors is a Poisson algebra.
The associative product is the usual -module product when one of the two factors has 0 degree. It is given by
for all and , if , , in which case
The Lie bracket is given by
when . It is given by(4.1)
is the operator given by
for all and , if , , in which case
is a surjective Poisson algebra map.
The proof is straightforward, but cumbersome. We have to prove associativity (and commutativity) of the product , and Jacobi identity and Leibniz rule for the bracket . The easiest case is when one of the involved V-multivectors has degree 0 and the rest have degree 1. We skip this case. The next to the easiest case is when all involved V-multivectors have degree 1. We only treat in some (but not all) details this case. The general case is similar and it is left to the reader. We begin by noticing that 1-V-multivectors are just derivations of V. Moreover, when , then is just the symbol of D, i.e. . Now let . Then
for all and . This shows that is a 2-V-multivector with
This holds similarly in higher degrees.
Next let . Then is the 3-V-multivector given by
for all and , where we also used (4.2). This clearly agrees with , showing associativity (in low degree). This holds similarly in higher degrees. Commutativity is obvious. The unit is the constant function 1.
We now pass to the bracket . When the entries have degree 1, the latter is just the commutator of derivations and the Jacobi identity is obvious in this case. For the Leibniz rule let , and , and compute
This concludes the proof. ∎
When V is a line bundle, the map embeds
into as an abelian Poisson subalgebra and an ideal.
There is also a notion of FWL V-multivector. In order to discuss it, it is useful to discuss derivations of pull-back vector bundles first. So, let V be a vector bundle, and consider its pull-back along a surjective submersion . Clearly, a derivation D of is completely determined by its symbol and its action on pull-back sections. The restriction of D to pull-back sections is a derivation along π, i.e. it is an -linear map and there exists a, necessarily unique, vector field along π, denoted by , fitting in the Leibniz rule
The correspondence establishes a -linear bijection between derivations D of and pairs consisting of a vector field and a derivation along π satisfying the following additional compatibility: . When is a vector bundle, it makes sense to talk about polynomial sections of . Namely, , where the tensor product is over , and multiplicative reals act on via their action on the first factor. As for functions, we denote by this action. A section v of is homogeneous (polynomial) of degree k if for all ; in other words, . Denote by the space of homogeneous sections of degree k, and by
the space of all polynomial sections. FWL sections are degree 1 sections and they identify with sections of . Core sections are degree 0 sections and they identify simply with sections of V. Similarly to vector fields, a derivation D of is homogeneous (polynomial) of degree k if it maps homogeneous sections of degree h to homogeneous sections of degree ; in other words,
for all . The space of all polynomial derivations of will be denoted by , and they correspond to pairs where and takes values in polynomial sections.
We can also consider polynomial symmetric -multivectors. We will only need core and FWL ones. A symmetric k--multivector D is FWL (resp. core) if it is homogeneous of degree (resp. ), i.e.
for all . We denote by (resp. the space of FWL (resp. core) symmetric -multivectors. The Lie bracket on V-multivectors preserves FWL ones. Additionally, the projection maps FWL V-multivectors to FWL multivector fields, and we get a short exact sequence of Lie algebras:
A k--multivector D is core if and only if
for all , , , and all .
The proof is straightforward. ∎
It easily follows from the above proposition that a core k--multivector is completely determined by its symbol. More precisely, the symbol map establishes a one-to-one correspondence between core k--multivectors and core multivector fields. Let us discuss this in detail in the case . The general case is similar. So, let D be a core 1--multivector. In other words, D is a degree -1 derivation of . From Proposition 4.4 it maps linear sections to core sections, and core sections to 0. Now let f be an FWL function on E and let . Then v and can be seen as a core and an FWL section of , respectively. Hence, we have
with . As the action of D on linear sections determines D completely, we see that D is necessarily of the form
for some core vector field on E. In particular, , the vertical lift of some section . Conversely, every derivation of of the type , with , is a core derivation. We conclude that core 1--multivectors identify with core vector fields on E via the symbol map: . Similarly,
In particular, does not really depend on V (but only on E).
A symmetric k--multivector D is FWL if and only if
for all , , , and all .
The proof is straightforward. ∎
In particular, an FWL symmetric k--multivector D determines a map
for all and all . The map is -multilinear and symmetric. Hence, it can be seen as a vector bundle map or, equivalently, as a section of .
The assignment establishes a -linear bijection between FWL symmetric k--multivectors and pairs consisting of an FWL symmetric multivector field and a vector bundle map such that .
The proof is easy and left to the reader. ∎
According to Proposition 4.6 we will sometimes call the pair itself an FWL symmetric -multivector.
We only need to explain the map I. To do that, we first remark that π-vertical vector fields act naturally on sections of , via
for all , and . Now, take and . Then
Equivalently, we can interpret as a core -multivector, via the isomorphism
and then multiply by the FWL function to get an FWL -multivector.
We conclude this section showing that FWL symmetric -multivectors do also identify with polynomial derivations of . This is an easy consequence (among other things) of Proposition 4.6. Indeed, take and let be the corresponding pair. Denote by the projection. We claim that can be seen as a derivation along π. Indeed, is a section of and, by acting on a section with the DL-factor, we get a section of , i.e. a polynomial section of . In the following, we use this construction to interpret as a derivation along π. If we do so, the pair consists of a vector field on , and a derivation along π, with the additional property that . Hence, it corresponds to a (polynomial) derivation of the pull-back vector bundle (recall from Section 3 that is the vector field on completely determined by the property of acting as on polynomial functions on , or, equivalently, core symmetric multivector fields on E, where is the Poisson bracket of symmetric multivector fields). Finally, a tedious but straightforward computation shows that the bijection between linear -multivectors and polynomial derivations of obtained in this way also preserves the Lie algebra structures. When we equip with the product by a core -multivector, the latter bijection becomes an isomorphism of Lie–Rinehart algebras. We have thus proved the following main result in this section.
Let and be vector bundles. The assignment establishes a degree inverting isomorphism of Lie–Rinehart algebras over between linear -multivectors and polynomial derivations of .
5 Differential operators and their symbols
We finally come to the object of our primary interest: differential operators. This section is a super-short review of the subject.
Let be vector bundles. A (linear) differential operator (DO in the following) of order q from V to W is an -linear map
for all . In particular, DOs of order zero are just vector bundle maps . We denote by the space of order q DOs from V to W. Clearly, a DO of order q is also a DO of order , and we get the filtration
The union of all will be denoted simply by . A scalar DO on M is a DO acting on functions over M, i.e. a DO from the trivial line bundle to itself. We use the symbol (instead of ) for scalar DOs.
The composition of an order q and an order r DO is an order DO. In particular, for all q, is a -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 is a filtered non-commutative algebra with the composition. It is actually the universal enveloping algebra of the tangent Lie algebroid . Being an associative algebra, 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 scalar DO.
Given an order q DO from V to W, and functions , the nested commutator
is an order 0 DO. Additionally, it is a derivation in each of the arguments and it is symmetric in those arguments. In this way, we get a map
The map σ is called the symbol and it fits in a short exact sequence of -modules
where the second arrow is the inclusion.
Vector fields are first order scalar DOs. Derivations of the vector bundle V are first order DOs D from V to itself such that belongs to . Additionally, we have .
For scalar DOs, the short exact sequence (5.1) becomes
The symbol of scalar DOs intertwines the commutator with the Poisson bracket (of symmetric multivector fields) in the sense that
whenever and , in which case we take .
We conclude this short review section by 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 , , be variables. A length k multi-index I is a word , with , where words are considered modulo permutations of their letters. The length k of a multi-index is also denoted by . 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 generated by . The length is then a monoid homomorphism. A length k multi-index , determines an order k DO
Now, we go back to manifolds M (and vector bundles over them). Actually, (likewise ) is the -module of sections of a vector bundle over M. If are coordinates on M, then is spanned locally (in the corresponding coordinate neighborhood) by
More precisely, locally, every DO can be uniquely written in the form
where the are local functions on M. The can be recovered via formulas
where, for a multi-index I, we denote by the product where is the number of times the letter i occurs in I.
Finally, if Δ is an order q scalar DO locally given by (5.2), then its symbol is locally given by
6 Core and fiberwise 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 an FWL DO is an FWL multivector field. It is also motivated by the linearization construction discussed in the next section. Yet another motivating little fact is that the Laplacian of an FWL metric is an FWL DO (Example 6.2).
Let be a vector bundle. We have learnt from Sections 2–4 that, given a type of geometric structures on manifolds (functions, vector fields, tensors etc.) appropriate notions of core and FWL structures of the type on E exist, and these notions can be identified by means of the following recipe: (1) notice that the space of structures of type on E possesses a subspace which is naturally graded (via the action of multiplicative reals on E by fiberwise scalar multiplication); (2) identify the smallest degree k for which the degree k homogeneous component of is non-trivial; (3) put and . A quick check shows that this recipe cooks up the required definitions in all cases considered so far. Notice that we could make this recipe much more rigorous using for the rather vague “geometric structure of type ” the very precise notion of natural vector bundle , 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 non-commutative algebra of scalar DOs . We begin noticing that, for each q, the space of DOs of order q possesses a subspace which is naturally graded:
where consists of degree k DOs (of order q), i.e. DOs Δ such that
for all . The smallest degree k for which is non-trivial is . So, following our recipe, we put
and call them core DOs. We also put
Let be vector bundle coordinates on E, and let be dual coordinates on . A DO is a core DO if and only if, locally,
where is a length q multi-index.
It follows from (6.2) that is the subalgebra generated by core functions and core vector fields . Equivalently, it is the universal enveloping algebra of the abelian Lie algebroid . Because of the latter description, there is an algebra isomorphism
mapping a monomial
to the DO
In its turn, as already mentioned, identifies with polynomial functions on . In the following, we will often identify with both and via the latter isomorphisms. If is locally given by (6.1), then it identifies with
where, for , we denote by the monomial .
We will always consider as a -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
DOs in are called fiberwise linear differential operators (FWL DOs) of order q.
It is also clear that for all q. More precisely,
A DO is FWL if and only if, in vector bundle coordinates, it looks like
It is easy to see from this formula that is the -submodule spanned by 1, and .
Let g be a metric on E, and assume it is FWL. Then the associated Laplacian operator
is an 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 is FWL as well. Now, the covariant derivative of a 1-form ϑ along the Levi-Civita connection is the covariant 2-tensor given by the formula
where is the musical isomorphism. Equivalently, the covariant derivative of a vector field Y along another vector field X is the vector field that acts on functions as follows:
where is the gradient of f. By 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 for all . As the Laplacian of a function f is obtained by contracting the covariant derivative of df with , decreases by one the degree of a homogeneous (fiberwise polynomial) function. So it is a second order DO of degree , i.e. an FWL DO of order 2, as claimed. It might also be interesting to remark that the Levi-Civita connection of an FWL metric is an FWL connection according to a definition introduced in .
The space of FWL DOs is the stabilizer of in the Lie algebra , i.e. a DO is in if and only if for all .
The “only if part” of the statement immediately follows from an obvious order/degree argument. For the “if part”, consider a DO Δ of order r. Locally,
Assume that Δ is in the stabilizer of . We want to show that Δ is the sum of operators of the form (6.3) (with possibly varying ). As is a core function for all i, the commutator is a core DO. But
so it can only be a core DO for all i if for , and . In other words, Δ is necessarily of the form
To conclude, it is enough to prove that is of the form . To do this, recall that is a core vector field for all α, and hence is a core DO. But, from (6.6),
which is a core DO for all α if and only if
for all , i.e. is a (non-necessarily homogeneous) first order polynomial in the variables u, as desired. ∎
It follows from Lemma 6.3 that is a Lie subalgebra in . As already mentioned, it is also a -submodule. Actually, it is a Lie–Rinehart algebra over , the anchor being the adjoint operator . To see this, first notice that is indeed a well-defined derivation of for all . Now, take and , and compute
Finally, from , we see that, for every , the derivation determines a polynomial vector field (of the same degree) on , also denoted by .
The sequence of Lie–Rinehart algebras
First of all, as already remarked, is in . Even more, as it is an abelian subalgebra in , it is actually in the kernel of . To see that core DOs exhaust the kernel of (i.e. is its own centralizer), assume that for all . Then exactly the same computation as in the proof of Lemma 6.3 shows that Δ is locally of the form (6.3) with , i.e. . For the exactness of the sequence (6.7) it remains to show that the map is surjective. To do that, we work in local coordinates again. So, let be vector bundle coordinates on E, and let be dual coordinates on . It is not hard to see that, if Δ is locally given by (6.3), then the vector field is locally given by
where, for a multi-index , we denote by the monomial (). As (6.8) is the local expression of a generic homogeneous polynomial vector field of degree , we are done. ∎
Our next aim is to prove that the Lie–Rinehart algebra is canonically isomorphic to the Lie–Rinehart algebra of polynomial derivations of an appropriate line bundle on . We begin with a simple proposition.
The symbol of an FWL DO is an FWL symmetric q-multivector field on E. Every FWL symmetric q-multivector field is the symbol of an order q FWL DO.
Now let . Notice that the adjoint operator , seen as a polynomial vector field on , corresponds exactly to the symbol via the isomorphism . It is also clear that, in view of its coordinate form (6.3), Δ is completely determined by or, equivalently, , together with the map
The map is clearly well-defined. Additionally, it enjoys the following properties:
is a first order DO in each entry.
More precisely, we have the following lemma.
The map satisfies
for all and .
Let 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. ∎
The data (determines Δ completely and) can be repackaged in a very useful way. Namely, consider the line bundle
Then the pair determines an FWL q--multivector in the following way. Recall that an FWL q--multivector can be equivalently presented as a pair consisting of an FWL symmetric multivector field and a vector bundle map such that . We claim that we can construct such a pair from the pair . Namely, we put
and define by putting
for all and . Equation (6.9) needs some explanations. In the first summand of the right-hand side, we interpret as a q-multiderivation of , so, when contracting it with the sections , we get a plain derivation . As already remarked, derivations of E act on the exterior algebra of E. In our case we have
for all and all .
The next theorem is the main result of the paper.
The assignment establishes a degree inverting isomorphism of Lie–Rinehart algebras .
First of all, we have to show that is well-defined, i.e. it is symmetric and -linear in all its arguments. The symmetry is obvious. For the linearity, let , and compute
Let us compute the two summands separately. First of all, for every ,
where is the section implicitly defined by
for all . We remark for future use that, in particular,
The endomorphism acts on E via its dual, which is minus its transpose, and hence
We also have
Putting everything together, we find
We conclude that is a vector bundle map as desired. Additionally, the composition does clearly agree with , so that is indeed a q--multivector. It is also clear that can be reconstructed from showing that the correspondence is injective. Next we prove the -linearity. So, take
so that . We want to show that . To do this, we begin by noticing that the product is the pair where is the symmetric multivector field that, when interpreted as a multiderivation , is given by
and, similarly, is the bundle map given by
for all . From the properties of the symbol map, we have , and it remains to take care of . So choose , and compute . From symmetry, it is enough to choose for all i and some φ. First of all, we have
Only the terms with (and hence , respectively) survive, and we get
Now, for all ,
We already computed the second summand, while the first summand is
The surjectivity of the map now follows from (local) dimension counting.
It remains to check that the isomorphism defined in this way is both anchor and bracket preserving. For the anchor, the anchor of is the derivation of
corresponding to the FWL multivector field , which is exactly .
For the bracket, as we already discussed -linearity and compatibility with the anchor, it is enough to discuss the brackets of generators. As already remarked, is generated (over ) by 1, and . A direct check shows that
for all and all . Here and are, respectively, the linear vector field on and the derivation of 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 from Theorem 4.7 (in the case ), we get a (degree inverting) Lie–Rinehart algebra isomorphism
that we denote by A again.
Let be vector bundle coordinates on E, and let be dual coordinates on . Denote by the local coordinate generator of . It is easy to check using, e.g., (6.8), (6.15) and the -linearity, that, if the operator is locally given by (6.3), then the corresponding derivation maps a local section of to
7 Linearization of differential operators
Let be a manifold, let be a submanifold and let be a scalar DO. Denote by the normal bundle to M, i.e. . In this section, we show that, under appropriate linearizability conditions, the DO Δ can be linearized around M yielding an FWL differential operator . The DO represents the first order approximation of Δ around M in the direction transverse to M. This linearization construction is a further motivation supporting our definition of FWL DOs.
So, let be a submanifold and let be its normal bundle. We will often consider adapted coordinates on around points of M, i.e. coordinates such that . In particular, the restrictions are coordinates on M. From we see that are conormal 1-forms and are vector bundle coordinates on E.
We want to explain what it means to linearize an order q DO operator around M. We proceed as follows: first, we recall the linearization of a function, second we discuss the linearization of a symmetric multivector field, and finally we define the linearization of a generic DO. So, let . We say that F is linearizable (around M) if . In this case, is a conormal 1-form to M, i.e. a section of the conormal bundle . Hence, it corresponds to an FWL function on E. We put and call it the linearization of F. For instance, if are adapted coordinates on , then the are linearizable and the linear fiber coordinates on E are their linearizations. If F is any linearizable function on , then locally, around a point of M, for some functions of the (given by ), and, in this case, . Notice that every linear function is the linearization of a (non-unique) linearizable function : .
We now pass to symmetric multivector fields. So, let P be a symmetric q-multivector field on . We say that P is linearizable if it belongs to the ideal in generated by vector fields that are tangent to M. In other words, M is a coisotropic submanifold of with respect to P.
Let be a linearizable symmetric multivector field on . Then there exists a unique FWL symmetric q-multivector field on E such that
for all linearizable functions . The linearization preserves the Poisson bracket of symmetric multivector fields.
We begin by remarking that, as , the function is clearly linearizable for any choice of linearizable functions . Now we want to show that the right-hand side of (7.1) does only depend on . We do this in coordinates. So, let be adapted coordinates on , and let be the associated vector bundle coordinates on E. Locally
It follows from that . Now compute
Using that and that , we find
which does only depend on the . As every FWL function is a linearization, this also shows that is well-defined on linear functions. Finally, can be uniquely extended to all functions on E, as a symmetric q-multivector field, also denoted by , and locally given by
In particular, is an FWL multivector field.
For the last part of the statement, first notice that the ideal is preserved by the Poisson bracket. So, if , then it makes sense to linearize . The rest follows easily from equation (7.1). ∎
Proposition 7.1 is a “symmetric multivector analogue” of the following well-known fact. Let be a Poisson manifold, let be its cotangent algebroid and let be a coisotropic submanifold. Then the conormal bundle is a subalgebroid , and hence the normal bundle NM is equipped with a linear Poisson structure (see, e.g., [14, Section 3]).
By definition, the multivector field in Proposition 7.1 is the linearization of P.
We finally come to a generic differential operator .
An order q DO is order q linearizable (around M) if its symbol is linearizable.
An order DO is always order q linearizable, but it might not be order linearizable.
A function is order 0 linearizable if and only if , i.e. F is a linearizable function.
A vector field is order 1 linearizable if and only if it is tangent to M, i.e. it is a linearizable vector field.
Now, let be an order q linearizable DO. In order to linearize it, we use Theorem 6.7. In other words, from Δ we cook up a q--multivector of , with as in Section 6. The construction is inspired by the proof of Theorem 6.7 itself. We let be the linearization of the symbol of Δ. It remains to define the vector bundle map
For , we define as follows. Pick linearizable functions such that and consider the following function on M:
for all . On the right-hand side, we interpret as a multiderivation of . We have to show that Φ is well defined, i.e. does only depend on the , and Φ is indeed a vector bundle map as desired, i.e. it is (symmetric and) -linear in its arguments. To see that does not depend on the choice of the , assume that . Hence, we have and . Additionally,
Now, is a second order DO. As the first jet of vanishes on M, the expression does only depend on the symbol of (and the Hessian of ). But
So, if is locally given by (7.2), using , we find that, locally,
We conclude that is well-defined. It remains to see that Φ is -linear in one, and hence in all, of its arguments. So, take . The simple formula
shows that, when we replace with in , we can replace with , where is any function such that . So, we have
For the second summand, compute
Now, to conclude the proof of the -multilinearity, it is enough to show that
But this follows easily from the fact that is the linearization of and formula (7.3) again.
Finally, by construction, , so the pair corresponds to a q--multivector. We call the linear DO the linearization of Δ and denote it .
The above discussion leads to the following theorem.
The assignment is a well-defined (linearization) map from order q linearizable DOs on and order q FWL DOs on E. The linearization preserves the commutator of DOs in the following sense: let and be an order linearizable and an order linearizable DO, respectively. Then is order linearizable, and its linearization is .
We only need to prove the last part of the statement. To do that, recall that the commutator is a DO of order and its symbol is the Poisson bracket . Let and be their linearizations. As the ideal in is preserved by the Poisson bracket, is order linearizable as desired. Let be its linearization. As the linearization of multivector fields preserves the Poisson bracket, . Finally, we have to take care of . It is a straightforward computation, which we sketch to point out the possible subtleties related to the properties of the various objects involved. Recall from the discussion preceding the statement of the theorem that
and likewise for . Here is any linearizable function on such that . In the following, to stress that, actually, the function does only depend on the , we write (instead of ). Additionally, we call f (resp. ) the f-component of the FWL DO (resp. ). We have to show that the f-component of agrees with . To do this, we compute explicitly. By symmetry and -multilinearity, it is enough to evaluate it on equal sections . So, let be a linearizable function such that . We have
The only terms that survive are those with (and hence , respectively). We call them , respectively. The first one is given by
Similarly, the third one is
To compute , we use a simple trick: for every two scalar DOs we have
After using this formula, we get
Now we use that both Δ and F are linearizable to replace with its restriction to M in the first summand (likewise for Δ in the second summand). We get
where we also used (7.4). At this point, it can be checked directly, exploiting the explicit formula (4.1) for the Poisson bracket of -multivectors, that agrees with the f-component of . We conclude that , as desired. ∎
Let be the total space of a vector bundle and interpret M as a submanifold in via the zero section. In this situation, the normal bundle NM to M identifies canonically with E itself. Clearly, an order q FWL DO is automatically order q linearizable, and it easily follows from the proof of Theorem 7.5 that the vector bundle isomorphism identifies Δ with its own linearization .
As mentioned in Section 1, the results in the present paper should be considered as a first step towards a theory of multiplicative/infinitesimally multiplicative DOs on Lie groupoids/Lie algebroids and their Lie theory. The reader may consult, e.g.,  for a recent review on multiplicative structures on Lie groupoids and their infinitesimal counterparts: infinitesimally multiplicative structures on Lie algebroids. We speculate that the ultimate definition of a multiplicative DO on a Lie groupoid (which we do not know yet) will fit into a theorem of the following form: Let be a Lie groupoid with Lie algebroid and let be a multiplicative DO on . Then Δ is linearizable around M and its linearization is an infinitesimally multiplicative DO on A; if is source simply connected, then this correspondence between multiplicative and infinitesimally multiplicative DO can be inverted. Finally, we expect that non-trivial higher order examples will come from the theory of multiplicative metrics on Lie groupoids. Namely, we speculate that the Laplacian operator (on functions) determined by a multiplicative metric on a Lie groupoid is a multiplicative DO, and hence its linearization is an infinitesimally multiplicative DO on the Lie algebroid of A. Exploring this class of examples requires a lot of work, including finding the appropriate definition of multiplicative/infinitesimally multiplicative DO and this goes far beyond the scopes of the present paper.
L. Vitagliano is a member of the GNSAGA of INdAM. We thank the referee for numerous suggestions that improved the presentation.
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