The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs

In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional manifolds. We show that the infinite jet space of the fiber bundle is a profinite dimensional manifold in a natural way. The formal solution space of the nonlinear PDE then is a subspace of this jet space, and inherits from it the structure of a profinite dimensional manifold, if the PDE is formally integrable. We apply our concept to scalar PDEs and prove a new criterion for formal integrability of such PDEs. In particular, this result entails that the Euler-Lagrange equation of a relativistic scalar field with a polynomial self-interaction is formally integrable.


Introduction
Even though it appears to be unsolvable in general, the problem to describe the moduli space of solutions of a particular nonlinear PDE has lead to powerful new results in geometric analysis and mathematical physics. Notably this can be seen, for example, by the fundamental work on the structure of the moduli space of Yang-Mills equations [5,12,34]. Among the many challenging problems which arise when studying moduli spaces of solutions of nonlinear PDE's is that the space under consideration does in general not have a manifold structure, usually not even one modelled on an infinite dimensional Hilbert or Banach space. Moreover, the solution space can possess singularities. A way out of this dilemma is to study compactifications of the moduli space like the completion of the moduli space with respect to a certain Sobolev metric, cf. [15]. Another way, and that is the one we are advocating in this article, is to consider a "coarse" moduli space consisting of so-called formal solutions of a PDE, i.e. the space of those smooth functions whose power series expansion at each point solves the PDE. In case the PDE is formally integrable in a sense defined in this article, the formal solution space turns out to be a profinite dimensional manifold. These possibly infinite dimensional spaces are ringed spaces which can be regarded as projective limits of projective systems of finite dimensional manifolds.
Profinite dimensional manifolds appear naturally in several areas of mathematics, in particular in deformation quantization, see for example [27], the structure theory of Lie-projective groups [6,21], in connection with functional integration on spaces of connections [4], and in the secondary calculus invented by Vinogradov [36,22] which inspired the approach in this paper.
The paper consists of two main parts. The first, Section 2, lays out the foundations of the theory of profinite dimensional manifolds. Besides the work [1], which is taylored towards explaining the differential calculus by Ashtekar and Lewandowski [4], literature on profinite dimensional manifolds is scarce. Moreover, our approach to profinite dimensional manifolds is novel in the sense that we define them as ringed spaces together with a so-called pfd structure, which consists not only of one but a whole equivalence class of representations by projective systems of finite dimensional manifolds. The major point hereby is that all the projective systems appearing in the pfd structure induce the same structure sheaf, which allows to define differential geometric concepts depending only on the pfd structure and not a particular representative. One way to construct differential geometric objects is by dualizing projective limits of manifolds to injective limits of, for example, differential forms, and then sheafify the thus obtain presheaves of "local" objects. Again, it is crucial to observe that the thus obtained sheaves are independant of the particular choice of a representative within the pfd structure, whereas the "local" objects obtain a filtration which depends on the choice of a particular representative. Using variants of this approach or directly the structure sheaf of smooth functions, we introduce in Section 2 tangent bundles of profinite dimensional manifolds and their higher tensor powers, vector fields, and differential forms.
The second main part is Section 3, where we introduce the formal solution space of a nonlinear PDE. We first explain the necessary concepts from jet bundle theory and on prolongations of PDE's in fiber bundles, following essentially Goldschmidt [19], cf. also [28,36,37,22]. In Section 3.2.2 we introduce in the jet bundle setting a notion of an operator symbol of a nonlinear PDE such that, in the linear case, it coincides with the well-known (principal) symbol of a partial differential operator up to canonical isomorphisms. The corresponding result, Theorem 3.17, appears to be new. Afterwards, we show that the bundle of infinite jets is a profinite dimensional manifold. This result immediately entails that the formal solution space of a formally integrable PDE is a profinite dimensional submanifold of the infinite jet bundle. Finally, in Section 3.4, we consider scalar PDE's. We prove there a widely applicable criterion for the formal integrability of scalar PDE's, which to our knowledge has not appeared in the mathematical literature yet. Moreover, we conclude from our criterion that the Euler-Lagrange Equation of a scalar field with a polynomial self-interaction on an arbitrary Lorentzian manifold is formally integrable, so its formal solution space is a profinite dimensional manifold. We expect that this observation will have fundamental consequences for a mathematically rigorous formulation of the quantization theory of such scalar fields.
Acknowledgements: The first named author (B.G.) is indebted to Werner Seiler for many discussions on jet bundles, and would also like to thank B. Kruglikov and A.D. Lewis for helpful discussions. B.G. has been financially supported by the SFB 647: Raum-Zeit-Materie, and would like to thank the University of Colorado at Boulder for its hospitality. The second named author (M.P.) has been partially supported by NSF grant DMS 1105670 and would like to thank Humboldt-University, Berlin for its hospitality.

Some notation
Let us introduce some notation and conventions which will be used throughout the paper.
If nothing else is said, all manifolds and corresponding concepts, such as submersions, bundles etc., are understood to be smooth and finite dimensional. The symbol T k,l stands for the functor of k-times contravariant and l-times covariant tensors, where as usual T := T 1,0 and T * := T 0,1 . If X is a manifold, then the corresponding tensor bundles will be denoted by π T k,l X : T k,l X → X. Moreover, we write X ∞ and Ω k for the sheaves of smooth vector fields and of smooth k-forms, respectively.
Given a fibered manifold, i.e. a surjective submersion π : E → X, we write Γ ∞ (π) for the sheaf of smooth sections of π. Its space of sections over an open U ⊂ X will be denoted by Γ ∞ (U; π). The set of local smooth sections of π around a point p ∈ M is the set of smooth sections defined on some open neighborhood of p and will be denoted by Γ ∞ (p; π). The stalk at p then is a quotient space of Γ ∞ (p; π) and is written as Γ ∞ p (π). The vertical vector bundle corresponding to the fibered manifold π is defined as the subvector bundle (1.1) π V : V(π) := ker(Tπ) −→ E of π TE : TE → E. If π ′ : E ′ → X is a second fibered manifold, the vertical morphism corresponding to a morphism h : E → E ′ of fibered manifolds over X is given by If π : E → X is a vector bundle, then the fibers of π are R-vector spaces, hence one can apply tensor functors fiberwise to obtain the corresponding tensor bundles. In particular, π ⊙ k : Sym k (π) → X will stand for the k-fold symmetric tensor product bundle of π.
Finally, unless otherwise stated, the notions "projective system" and "projective limit" will always be understood in the category of topological spaces, where they of course exist; see [14,Chap. VIII,Sec. 3]. In fact, given such a projective system M i , µ ij i,j∈N,i≤j , a distinguished projective limit is given as follows. Define M := (p i ) i∈N ∈ i∈N M i | µ ij (p j ) = p i for all i, j ∈ N with i ≤ j to be the subspace of all threads in the product, and the continuous maps µ i : M → M i as the restrictions of the canonical projections i∈N M i → M i to M. Then one obviously has µ ij • µ j = µ i for all i, j ∈ N with i ≤ j. Note that a basis of the topology of M is given by the set of all open sets of the form µ −1 i (U), where i ∈ N and U ⊂ M i is open. In the following, we will refer to the thus defined M together with the maps (µ i ) i∈N as the canonical projective limit of M i , µ ij i,j∈N,i≤j , and denote it by M = lim ←− i∈N M i .

Profinite dimensional manifolds
In this section, we introduce the concept of profinite dimensional manifolds and establish the differential geometric foundations of this new category.
2.1. The category of profinite dimensional manifolds. The following definition lies in the center of the paper: Definition 2.1. a) By a smooth projective system we understand a family M i , µ ij i,j∈N, i≤j of smooth manifolds M i and surjective submersions µ ij : M j → M i for i ≤ j such that the following conditions hold true: a , µ ′ ab a,b∈N, a≤b denotes a second smooth projective system, a morphism of smooth projective systems between M i , µ ij i,j∈N, i≤j and M ′ a , µ ′ ab a,b∈N, a≤b is a pair ϕ, (F a ) a∈N consisting of a strictly increasing map ϕ : N → N and a family of smooth maps F a : M ϕ(a) → M ′ a , a ∈ N such that for each pair a, b ∈ N with a ≤ b the diagram commutes. We usually denote a smooth projective system shortly by M i , µ ij and write to indicate that ϕ, (F a ) a∈N is a morphism of smooth projective systems. If each of the maps F a is a submersion (resp. immersion), we call the morphism ϕ, F a a submersion (resp. immersion).
c) Two smooth projective systems M i , µ ij and M ′ a , µ ′ ab are called equivalent, if there are surjective submersions A pair of such surjective submersions will be called an equivalence transformation of smooth projective systems.

Remark 2.2.
In the definition of smooth projective systems and later in the one of smooth projective representations we use the partially ordered set N as index set. Obviously, N can be replaced there by any partially ordered set canonically isomorphic to N such as an infinite subset of Z bounded from below. We will silently use this observation in later applications for convenience of notation.
Example 2.3. a) Let M be a manifold. Then M i , µ ij with M i := M and µ ij := id M for i ≤ j is a smooth projective system which we call trivial and which we denote shortly by M, id M . b) Assume that for i ≤ j one has given surjective linear maps λ ij : V j → V i between real finite dimensional vector spaces such that (SPS1) and (SPS2) are satisfied. Then V i , λ ij is a smooth projective system. For example, this situation arises in deformation quantization of symplectic manifolds when constructing the completed symmetric tensor algebra of a finite dimensional real vector space; see [27] for details. Of course, a simpler example is given by the canonical projections π ij : R j → R i onto the first i coordinates, hence R i , π ij is a (non-trivial) smooth projective system. c) In the structure theory of topological groups [21,6] one considers smooth projective systems (G i , η ij ) such that each G j is a Lie Group and the η ij : G j → G i are continuous group homomorphisms. See Example 2.8 c) below for a precise description of the projective limits of such projective systems of Lie groups. d) The tower of k-jets over a fiber bundle together with their canonical projections forms a smooth projective system (see Section 3.1).
Within the category of (smooth finite dimensional) manifolds, a projective limit of a smooth projective system obviously does in general not exist. In the following, we will enlarge the category of manifolds by the so-called profinite dimensional manifolds (and appropriate morphisms). The thus obtained category will contain projective limits of smooth projective systems. Definition 2.4. a) By a smooth projective representation of a commutative locally R-ringed space (M, C ∞ M ) we understand a smooth projective system M i , µ ij together with a family of continuous maps µ i : M → M i , i ∈ N, such that the following conditions hold true: (PFM1) As a topological space, M together with the family of maps We usually denote a smooth projective representation briefly as a family M i , µ ij , µ i . b) A smooth projective representation M i , µ ij , µ i of (M, C ∞ M ) is said to be regular, if each of the maps µ ij : M j → M i is a fiber bundle. c) Two smooth projective representations M i , µ ij , µ i and M ′ a , µ ′ ab , µ ′ a of (M, C ∞ M ) are called equivalent, if there is an equivalence transformation of smooth projective systems and µ ′ a = F a • µ ϕ(a) for all i, a ∈ N. In the following, we will sometimes call such a pair of surjective submersions an equivalence transformation of smooth projective representations. The equivalence class of a smooth projective system M i , µ ij , µ i will be simply denoted by [ M i , µ ij , µ i ] and called a pfd structure on (M, C ∞ M ). Proposition 2.5. Let (M, C ∞ M ) be a commutative locally R-ringed space with a smooth projective representation M i , µ ij , µ i . Assume further that M ′ a , µ ′ ab is a smooth projective system which is equivalent to M i , µ ij . Then there are continuous maps µ ′ a : M → M ′ a , a ∈ N, such that M ′ a , µ ′ ab , µ ′ a becomes a smooth projective representation of (M, C ∞ M ) which is eqivalent to M i , µ ij , µ i .
Proof. Choose an equivalence transformation of smooth projective systems Let us show first that M together with the family of continuous maps µ ′ a , a ∈ N is a projective limit of M ′ a , µ ′ ab . So assume that X is a topological space, and h a : Since M is a projective limit of M i , µ ij , this entails h = h. This proves that M is a projective limit of M ′ a , µ ′ ab . Next let us show that (PFM2) holds true with the µ i replaced by the µ ′ a . So let U ⊂ M be open, f ∈ C ∞ M (M), and p ∈ U. Choose i ∈ N such that there is an open U i ⊂ M I and a smooth f i : Then f a is smooth, and Finally, it remains to prove that µ i = G i • µ ′ ψ(i) for all i ∈ N, but this follows from • µ ϕ(ψ(i))) = µ iϕ(ψ(i))) • µ ϕ(ψ(i))) = µ i . This finishes the proof.
. By definition, it is clear that the composition of smooth maps between profinite dimensional manifolds is smooth, hence profinite dimensional manifolds and the smooth maps between them as morphisms form a category, the isomorphisms of which can be safely called diffeomorphisms. All of this terminology is justified by the simple observation Example 2.8 a) below.
Example 2.8. a) Given a manifold M, the trivial smooth projective system M, id M defines a smooth projective representation for the ringed space (M, C ∞ M ). Hence, every manifold is a profinite dimensional manifold in a natural way, and the category of manifolds a full subcategory of the category of profinite dimensional manifolds. b) Assume that M i , µ ij is a smooth projective system. Let together with the natural projections µ i : M → M i denote the canonical projective limit of M i , µ ij . Then, (PFM1) is fulfilled by assumption, and it is immediate that M carries a uniquely determined structure sheaf C ∞ M which satisfies (PFM2). The locally ringed space (M, C ∞ M ) together with the pfd structure [ M i , µ ij , µ i ] then is a profinite dimensional manifold. This profinite dimensioal manifold is even a projective limit of the projective system M i , µ ij within the category of profinite dimensional manifolds. We therefore write in this situation and call (M, C ∞ M ) (together with [ M i , µ ij , µ i ]) the canonical smooth projective limit of M i , µ ij . c) A locally compact Hausdorff topological group G is called Lie projective, if every neighbourhood of the identity contains a compact Lie normal subgroup, i.e. a normal subgroup N ⊂ G such that G/N is a Lie group. One has the following structure theorem [6,Thm. 4.4], [21]. A locally compact metrizable group G is Lie projective, if and only if there is a smooth projective system (G i , η ij ) as in Example 2.3 c) together with continuous group homomorphisms η i : G → G i , i ∈ N such that (G, η i ) is a projective limit of (G i , η ij ). Again, it follows that G carries a uniquely determined structure sheaf C ∞ G satisfying (PFM2). The locally ringed space (G, C ∞ G ) together with the pfd structure [ G i , η ij , η i ] becomes a regular profinite dimensional manifold with a group structure such that all of its structure maps are smooth. d) The space of infinite jets over a fiber bundle canonically is a profinite dimensional manifold (see Section 3.3).
Remark 2.9. In the sequel, (M, C ∞ M ) or briefly M will always denote a profinite dimensional manifold. Moreover, M i , µ ij , µ i always stands for a smooth projective representation defining the pfd structure on M. The sheaf of smooth functions on a profinite dimensional manifold will often briefly be denoted by C ∞ , if no confusion can arise.
Let N ⊂ M be a subset, and assume further that for some smooth projective representation M i , µ ij , µ i of the pfd structure on M the following holds true: (PFSM1) There is a stricly increasing sequence (l i ) i∈N such that for every i ∈ N the set N i : Observe that the ν ij are surjective by definition of the manifolds N i and by ν i = ν ij • ν j , where we have put ν i := µ ′ l i |N . In particular, (N i , ν ij ) becomes a smooth projective system. Proof. We first show that N together with the maps ν i is a (topological) projective limit of the projective system (N i , ν ij ). Let p i ∈ N i , i ∈ N such that ν ij (p j ) = p i for all j ≥ i. Since M together with the µ i is a projective limit of (M i , µ ij ), there exists an p ∈ M such that µ l i (p) = p i for all i ∈ N. By axiom (PFSM2), p ∈ N, hence one concludes that N is a projective limit of the manifolds N i .
Next, we show that C ∞ |N coincides with the uniquely determined sheaf C ∞ N satisfying axiom (PFM2). Since the canonical embeddings N i ֒→ M l i are smooth by (PFSM1), the embedding N ֒→ M is smooth as well, and C ∞ |N is a subsheaf of the sheaf C ∞ N . It remains to prove that for every open V ⊂ N a function f ∈ C ∞ N (V ) is locally the restriction of a smooth function on M. To show this let p ∈ V and V i an open subset of some N i such that p ∈ ν −1 i (V i ) ⊂ V , and such that there is Since N i is locally closed in M l i , we can assume after possibly shrinking V i that there is an open , which proves that f ∈ C ∞ |N (V ). The claim follows.
and let B n (0) be the open unit ball in R n . The projective limit then becomes a profinite dimensional submanifold of R ∞ . Note that it is not locally closed in R ∞ .
c) The space of formal solutions of a formally integrable partial differential equation is a profinite dimensional submanifold of the space of infinite jets over the underlying fiber bundle (see Section 3.3).
We continue with: We denote the space of local functions over U by C ∞ loc (U). Remark 2.13. a) Observe that C ∞ loc forms a presheaf on M, which depends only on the pfd structure [ M i , µ ij , µ i ]. Moreover, it is clear by construction that for every open U ⊂ M and every representative (M i , µ ij , µ i ) of the pfd structure, C ∞ loc (U) together with the family of pull-back maps µ * i : C ∞ (µ i (U)) → C ∞ loc (U) is an injective limit of the injective system of linear spaces C ∞ (µ i (U)), µ * ij i∈N . b) C ∞ loc is in general not a sheaf unless M is a finite dimensional manifold. The sheaf associated to C ∞ loc naturally coincides with C ∞ since locally, every smooth function is local.
Observe that this filtration has the property that

Tangent bundles and vector fields.
The tangent space at a point of a finite dimensional manifold can be defined as a set of equivalence classes of germs of smooth paths at that point or as the space of derivations on the stalk of the sheaf of smooth functions at that point. The definition via paths can not be immediately carried over to the profinite dimensional case, so we use the derivation approach.
Definition 2.14. Given a point p of the profinite dimensional manifold M, the tangent space of M at p is defined as the space of derivations on C ∞ p , the stalk of smooth functions at p, i.e. as the space Elements of T p M will be called tangent vectors of M at p. The tangent bundle of M is the disjoint union Note that for every i ∈ N there is a canonical map Tµ i : TM → TM i which maps a tangent vector Y ∈ T p M to the tangent vector . By construction, one has Tµ ij • Tµ j = Tµ i for i ≤ j. We give TM the coarsest topology such that all the maps Tµ i , i ∈ N are continuous. Now we record the following observation: Lemma 2.15. The topological space TM together with the maps Tµ i is a projective limit of the projective system TM i , Tµ ij .
Proof. Assume that X is a topological space, and Φ i i∈N a family of continuous maps Φ i : X → TM i such that Tµ ij • Φ j = Φ i for all i ≤ j. Since M is a projective limit of the projective system M i , µ ij , there exists a uniquely determined continuous map ϕ : Now let x ∈ X, and put p := ϕ(x) and e. let f be a smooth function defined on a neighborhood U of p, and [f ] p its germ at p. Then there exists i ∈ N, an open neighborhood U i ⊂ M i of p i and a smooth function We have to show that Φ(x) is independant of the choices made, and that it is a derivation indeed. So let f ′ : Without loss of generality, we can assume i ≤ j.
By assumption . But this implies, using the assumption on the Φ i that p be two germs of smooth functions at p. Then, after possibly shrinking the domains of f and g, one can find an i ∈ N, an open neighborhood which means that Φ(x) is a derivation.
By construction, it is clear that Let us verify that Φ(x) is uniquely determined by this property. So Finally, we observe that Φ : X → TM is continuous, since all maps Φ i = Tµ i Φ are continuous, and TM carries the initial topology with respect to the maps Tµ i .
This concludes the proof that TM together with the maps Tµ i is a projective limit of the projective system TM i , Tµ ij . Remark 2.16. a) If p ∈ M, Y p , Z p ∈ T p M, and λ ∈ R, then the maps Y p + Z p : C ∞ p → R and λY p : C ∞ p → R are derivations again. Hence T p M becomes a topological vector space in a natural way and one has T p M ∼ = lim ←− i∈N T µ i (p) M i canonically as topological vector spaces. In particular, this implies that π TM : TM → M is a continuous family of vector spaces. Note that this family need not be locally trivial, in general. b) Denote by P ∞ M,p the set of germs of smooth paths γ : (R, 0) → (M, p). There is a canonical map P ∞ M,p → T p M which associates to each germ of a smooth path γ : . Unlike in the finite dimensional case, this map need not be surjective, in general. But note the following result.
Proposition 2.17. In case the profinite dimensional manifold M is regular, the "dot map" Proof. We start with an auxiliary construction. Choose a smooth projective representation (M i , µ ij , µ i ) within the pdf structure on M such that all µ ij are fiber bundles. Put p i := µ(p i ) for every i ∈ N. Then choose a relatively compact open neighborhood U 0 ⊂ M 0 of p 0 which is diffeoemorphic to an open ball in some R n . In particular, U 0 is contractible, hence the fiber bundle µ 01 |µ −1 be a trivialization of that fiber bundle, and D 1 ⊂ F 1 an open neighborhood of p 1 which is diffeomeorphic to an open ball in some euclidean space. Put U 1 := Ψ −1 0 (U 0 × D 1 ). Then, U 1 is diffeomeorphic to a ball in some euclidean space, and µ 01|U 1 : U 1 → U 0 is a trivial fiber bundle with fiber D 1 . Assume now that we have constructed U 0 ⊂ M 0 , . . . , U j ⊂ M j such that for all i ≤ j the following holds true: (1) the set U i is a relatively compact open neighborhood of p i diffeomorphic to an open ball in some euclidean space, fiber bundle with fiber D i diffeomorphic to an open ball in some euclidean space. Let us now construct U j+1 and D j+1 . To this end note first that Then, U j is diffeomeorphic to a ball in some euclidean space, and µ jj+1 |U j+1 : U j+1 → U j is a trivial fiber bundle with fiber D j+1 . This finishes the induction step, and we obtain U i ⊂ M i and D i such that the three conditions above are satisfied.
After these preliminaries, assume that Z ∈ T p M is a tangent vector.
To start, choose a smooth path γ 0 : This finishes the induction step, and we obtain a family of smooth paths γ i with the desired properties.
Since M is the smooth projective limit of the M i , there exists a uniquely determined smooth path γ : R → M such that µ i • γ = γ i for all i ∈ N. In particular, this entails γ(0) = p, andγ(0) = Z, or in other words that Z is in the image of the map P ∞ M,p → T p M. Let us define a structure sheaf C ∞ TM on TM. To this end call a continuous map f ∈ C (U) defined on an open set U ⊂ TM smooth, if for every tangent vector Z ∈ U there is an i ∈ N, an open neighborhood open then form the section spaces of a sheaf C ∞ TM which we call the sheaf of smooth functions on TM. By construction, the family TM i , Tµ ij , Tµ i now is a smooth projective representation of the locally ringed space TM, C ∞ TM , hence TM, C ∞ TM becomes a profinite dimensional manifold. Since µ i • π TM = π TM i • Tµ i for all i ∈ N, one immediatly checks that the canonical map π TM : TM → M is even a smooth map between profinite dimensional manifolds. With these preparations we can state: Proposition and Definition 2.18. The profinite dimensional manifold given by TM, C ∞ TM and the pfd structure [ TM i , Tµ ij , Tµ i ] is called the tangent bundle of M, and π TM : TM → M its canonical projection. The pfd structure Proof. In order to check the last statement, consider a smooth pro- Tµ ij such that the following diagrams commute for all i, a ∈ N: y y r r r r r r r r r r y y r r r r r r r r r r This entails that the smooth projective representations TM i , Tµ ij , Tµ i and TM ′ a , Tµ ′ ab , Tµ ′ a of the tangent bundle (TM, C ∞ TM ) are equivalent, and the proof is finished. Remark 2.19. a) By Example 2.8 b), the induced smooth projective system TM i , Tµ ij has the canonical smooth projective limit Denote its canonical maps by Tµ i : TM → TM i . By the universal property of projective limits there exists a unique smooth map By construction of the profinite dimensional manifold structure on the tangent bundle TM, the map τ is even a linear diffeomorphism, and is in fact given by ij into a projective system of (finite dimensional) real vector spaces. By Theorem A.4, its projective limit within the category of locally convex topological Hausdorff spaces is given by T k,0 p M together with the continuous linear maps T k,0 µ p,i , that means we have and give T k,0 M the coarsest topology such that all the canonical maps are continuous. By construction, T k,0 M together with the maps T k,0 µ i has to be a projective limit of the projective system T k,0 M i , T k,0 µ ij . The sheaf of smooth functions C ∞ T k,0 M is uniquely determined by requiring axiom (PFM2) to hold true. One thus obtains a profinite dimensional manifold which depends only on the equivalence class of the smooth projective representation and which will be denoted by T k,0 M in the following. Moreover, T k,0 even becomes a functor on the category of profinite dimensional manifolds. If (N, C ∞ N ) is another profinite dimensional manifold and f : M → N a smooth map, then one naturally obtains the smooth map We continue with: The space of smooth vector fields over U will be denoted by X ∞ (U).
Assume that for U ⊂ M open we are given a smooth vector field V : U → TM and a smooth function f : U → R. We then define a function V f over U by putting for p ∈ U Proof. Choose a point p ∈ U, and then an open U i ⊂ M i and a function Then g pj is smooth, hence , and V f is smooth indeed.
is R-linear. It remains to check that δ V is a derivation, or in other words that it satisfies Leibniz' rule. But this follows immediately by the definition of the action of V on C ∞ (U) and the fact that V (p) ∈ Der C ∞ p , R for all p ∈ U. More precisely, one has, for p ∈ U and f, g ∈ C ∞ (U), This finishes the proof.
The space of local vector fields over U will be denoted by X ∞ loc (U). Remark 2.24. a) Obviously, X ∞ is a sheaf of C ∞ -modules on M, and X ∞ loc a presheaf of C ∞ loc -modules. Note that X ∞ loc depends only on the pfd structure , and pick a representative (M i , µ ij , µ i ) of the underlying pfd structure. If (m i ) i∈N is a sequence of integers such that (2.10) holds true, we sometimes say that V is of type (m 0 , m 1 , m 2 , . . .) with respect to the smooth projective representation (M i , µ ij , µ i ). The notion of the type of a local vector field is known from jet bundle literature [2], where it makes perfekt sense, since the profinite dimensional manifold of infinite jets has a distinguished representative of the underlying pfd structure, see Section 3.3. Now we are in the position to prove the following structure theorem: holds true, which entails that V pi = Tµ ij • V pj . Hence, the sequence of tangent vectors (V pi ) i∈N defines an element V p in We thus obtain a section V : M → TM, p → V p . Let us show that V is smooth. To this end, consider the composition By construction V i (p) = V pi for all p ∈ M. It suffices to show that each of the maps V i is smooth. To show this, choose a coordinate neighborhood U i ⊂ M i of µ i (p), and coordinates is a local coordinate system of TM i . The map V i now is proven to be smooth, if dx l • V i is smooth for 1 ≤ l ≤ k. But . Hence each V i is smooth, and V is a smooth vector field on M which satisfies δ V = D. This proves surjectivity.
Injectivity: Assume that V is a smooth vector field on M such that δ V = 0. This means that δ V f (p) = 0 for all f ∈ C ∞ (M) and p ∈ M. Choose now a i ∈ N and let f i be a smooth function on M i . Put which implies that V i (p) = 0 for all p ∈ M. Since V (p) is the projective limit of the V i (p), we obtain V (p) = 0 for all p ∈ M, hence V = 0. This finishes the proof that δ is bijective.
Local vector fields: Next, let us show that for a local vector field V : M → TM the derivation δ V maps local functions to local ones. To this end choose for every i ∈ N an integer m i ≥ i such that there exists a smooth vector field V im i : M m i → TM i along µ im i which satisfies Invariance of C ∞ loc (M): Finally, we have to show that if δ V for V ∈ X ∞ (M) leaves the space C ∞ loc (M) invariant, the vector field V has to be local. To this end fix i ∈ N and choose a proper embedding Then χ l •µ i ∈ C ∞ loc (M) for l = 1, . . . , N, hence there exist by assumption j 1 , . . . , j N ∈ N and g il ∈ C ∞ (M j l ) such that After possibly increasing the j l , we can assume that m i := j 1 = . . . = j N ≥ i. Denote by z l : R n → R the canonical projection onto the l-th coordinate, and define the vector field V im i :

Since by construction
is well-defined and satisfies The following result is an immediate consequence of Theorem 2.25.
is a derivation on C ∞ (M). Its corresponding underlying vector field will be denoted by [V, W ] as well, and will be called the Lie bracket of V and W . The Lie bracket of vector fields turns X ∞ (M) into a Lie algebra.
More precisely, this means that for all y ∈ µ −1 i (U i ), and V 1 , . . . , V k ∈ π −1 (y) ⊂ TM the relation holds true. In particular, a k-form ω over U is antisymmetric and k-multilinear in its arguments. The space of k-forms over U will be denoted by Ω k (U) where here and from now on we use the notation µ * i ω i for the form ω i • T k,0 µ i . The space of local k-forms over U will be denoted by Ω k loc (U).
Remark 2.28. a) By a straightforward argument one checks that the spaces Ω k (U) and Ω k loc (U) only depend on the pfd structure [(M i , µ ij , µ i )]. Moreover, for every representative (M i , µ ij , µ i ) of the pfd structure, Ω k loc (U) together with the family of pull-back maps µ * i : Ω k (µ i (U)) → Ω k loc (U) is an injective limit of the injective system of linear spaces Ω k (µ i (U)), µ * ij i∈N . b) By construction, it is clear that Ω k forms a sheaf of C ∞ -modules on M and Ω k loc a presheaf of C ∞ loc -modules. Moreover, Ω k coincides with the sheaf associated to Ω k loc . c) The representative M := (M i , µ ij , µ j ) of the pfd structure on M leads to the particular filtration F M • of the presheaf Ω k loc of local kforms on M by putting, for l ∈ N, Observe that this filtration has the property that Proposition and Definition 2.29. a) There exists a uniquely determined morphism of sheaves d : Ω k → Ω k+1 such that The morphism d is called the exterior derivative, fulfills d • d = 0, and maps Ω k loc to Ω k+1 loc . b) There exists a uniquely determined morphism of sheaves ∧ : Ω k × Ω l −→ Ω k+l , called the wedge product, such that for all i ∈ N, The wedge product also leaves Ω • loc invariant. c) Given a vector field V ∈ X ∞ (M), there exists the contraction with V that means the sheaf morphism which is uniquely determined by the requirement that for all ω ∈ Ω k (U) with U ⊂ M open, p ∈ U, and W 1 , . . . , W k−1 ∈ T p M the relation holds true. If V is a local vector field, contraction with V leaves Ω • loc invariant.
Proof. Using the sheaf property of Ω k one can reduce the claims to local statements which are immediately proved.

Jet bundles and formal solutions of nonlinear PDE's
The aim of this section is to develop a precise geometric notion of formally integrable (systems of) partial differential equations, and to show that the formal solution spaces of these equations canonically become a profinite dimensional manifold in the sense of Section 2. Finally, we are going to give a criterion for the formal integrability of nonlinear scalar partial differential equations, and apply this result to a class of interacting relativistic scalar field theories that arise in theoretical physics.
We refer the reader to [31,18,22] and also to [28,32] for introductionary texts on jet bundles, where the latter two references have a strong focus on the highly nontrivial algorithmic aspects of this theory. A nice short overview is also included in the introduction of [37].

Finite order jet bundles.
For the rest of the paper, we fix a fiber bundle π : E → X. Moreover, F will denote the typical fiber of π and we set m := dim X, n := dim F . Then one has dim E = m + n and the fibers π −1 (p) ⊂ E become n-dimensional submanifolds, which are diffeomorphic to F . There are distinguished charts for E: Definition 3.1. A manifold chart (x, u) : W → R m × R n of E defined over some open W ⊂ E is called a fibered chart of π, if for all e, e ′ ∈ W with π(e) = π(e ′ ) the equality x(e) = x(e ′ ) holds true.
Remark 3.2. a) Sometimes, fibered charts are called adapted charts. b) Note that a fibered chart (x, u) : W → R m × R n for π canonically gives rise to a well-defined manifold chart on X. It is given bỹ where e ∈ W ∩ π −1 (p) is arbitrary. c) On the other hand, a manifold atlas for E that consists of fibered charts for π can be constructed from manifold charts for X and from the local triviality of E as follows: For an arbitrary e ∈ E, take a bundle chart φ : π −1 (U) → U × F around π(e), that is, U is an open neighbourhood of π(e) and φ : π −1 (U) → U × F is a diffeomorphism such that commutes. Letx : U → R m be a manifold chart of X (here we assume that U is small enough), and letũ : B → R n be a manifold chart of F . Then is a fibered chart of π. Note here that by the commutativity of (3.2), the notation "x" is consistent with (3.1).
Let us introduce the following notation for multi-indices, which will be convenient in the following: For any k 1 , k 2 ∈ N with k 1 ≤ k 2 let N m k 1 ,k 2 denote the set of all multi-indices I ∈ N m such that and let F(m, k 1 , k 2 ) denote the linear space of all maps N m k 1 ,k 2 → R. For any l ≤ m, i 1 , . . . , i l ∈ {1, . . . , m}, the symbol 1 i 1 ...i l ∈ N m will denote the multi-index which has a 1 in its i j 's slot for j = 1, . . . , l, and a 0 elsewhere.
Any ψ ∈ Γ ∞ (p; π) allows the following local description: Choose a fibered chart (x, u) : W → R m × R n of π with W ∩ π −1 (p) = ∅. Then one has x • ψ =x near p, so that ψ is determined by the coordinates (u 1 • ψ, . . . , u n • ψ) = u • ψ near p. The special form of the following definition is motivated by the latter fact: for all α = 1, . . . , n and all I ∈ N m 0,k . The corresponding equivalence class j k p ψ of ψ is called the k-jet of ψ at p. Let us now come to several structures that can be defined via jets. Denoting by the collection of all k-jets in π, we obtain the surjective maps Using these maps, one can give J k (π) the structure of a finite dimensional manifold in a canonical way: For every fibered chart (x, u) : W → R m × R n of π and every I ∈ N m 0,k , one defines the map The following result is checked straightforwardly (cf. [31]).
Proposition and Definition 3.5. The maps (3.6) define an m + n dim F(m, 0, k)-dimensional manifold structure on J k (π). In view of this fact, J k (π) is called the k-jet manifold corresponding to π.
For convenience, we set J 0 (π) := E and j 0 p ψ := ψ(p) for any ψ ∈ Γ ∞ (p; π), and π 0 := π. More generally, we have for any k 1 ≤ k 2 the smooth surjective maps which satisfy π k,k = id J k (π) , and if one also has k 2 ≤ k 3 , then the following diagram commutes: Let us collect all structures underlying the above maps. Let (x, u) : W → R m × R n be a fibered chart of π. Then we set (π k , u k ) : If π is a vector bundle, then, for every p ∈ X, the fiber π −1 k (p) canonically becomes a linear space through , then the fiber π −1 k−1,k (a) carries a canonical affine structure which is modelled on the linear space Sym k T * π k−1 (a) X ⊗ ker Tπ |π 0,k−1 (a) .

(3.10)
To see the latter fact, assume that π 0,k−1 (a) ∈ W , let j k π k−1 (a) ψ ∈ π −1 k−1,k (a) and let v be an element of (3.10). Then v can be uniquely expanded as where we have used the abbreviation so that one can define j k π k−1 (a) ψ + v ∈ π −1 k−1,k (a) to be the uniquely determined element whose image under (3.9) is given by With these preparations, one has: Lemma 3.6. Let k, k 1 , k 2 ∈ N with k 1 ≤ k 2 . Then the following assertions hold.
a) The maps (3.8) turn π k : J k (π) → X into a fiber bundle with typical fiber F(m, 0, k) n . If π is a vector bundle, then so is π k . b) The maps (3.9) turn π k 1 ,k 2 : J k 2 (π) → J k 1 (π) into a fiber bundle with typical fiber F(m, l + 1, k) n , and π k−1,k : J k (π) → J k−1 (π) becomes an affine bundle, modelled on the vector bundle Proof. The reader can find a detailed proof of Lemma 3.6 in Chapter 6 of [31].
We close this section with a simple observation about distinguished elements of Γ ∞ (π k ). Let U ⊂ X be an open subset for the moment. Then for any ψ ∈ Γ ∞ (U; π), the map U → J k (π), p → j k p ψ, defines an element of Γ ∞ (U; π k ), called the k-jet prolongation of ψ. In fact, this construction induces a morphism of sheaves j k : Γ ∞ (π) → Γ ∞ (π k ) (with values in the category of sets) such that It should be noted that it is not possible to write an arbitrary element of Γ ∞ (U; π k ) as j k U ψ for some ψ ∈ Γ ∞ (U; π). The elements of Γ ∞ (U; π k ) having the latter property are called projectable. This notion is motivated by the following simple observation which follows readily from (3.11).
3.2. Partial differential equations. The aim of this section is to give a precise global definition of partial differential equations and the solutions thereof in the setting of arbitrary fiber bundles. We shall first consider the general (possibly nonlinear) situation in Section 3.2.1. Then, in Section 3.2.2, we are going to relate everything with the corresponding classical linear concepts.
Throughout this section, let π : E → X be a second fiber bundle, with typical fiber F and fiber dimension n.

General facts. We start with:
Definition 3.8. a) A subset E ⊂ J k (π) is called a partial differential equation on π of order ≤ k, if π k|E : E → X is a fibered submanifold of π k . b) Let E ⊂ J k (π) be a partial differential equation on π of order ≤ k.
For an open U ⊂ X, a section ψ ∈ Γ ∞ (U; π) will simply be called a solution of E, if ψ is a solution of E in p for every p ∈ U, that is, if im(j k ψ) ⊂ E.
The point of this definition is that one has seperated and globalized the notions "partial differential equation", "solution of a partial differential equation" and "partial differential operator". We shall first clarify how the latter concept fits into definition 3.8 a). Definition 3.9. A morphism h : J k (π) → E of fibered manifolds over X is called a partial differential operator of order ≤ k from π to π.
Of course, the notion "operator" in definition 3.9 is justified by the fact that as a morphism of fibered manifolds, any h as in Definition 3.9 induces the morphism of set theoretic sheaves We define D k (π, π) to be the set of all partial differential operators of order ≤ k from π to π, and remark that the assignment h → P h induces an injection P • of D k (π, π) into the set theoretic sheaf morphisms Γ ∞ (π) → Γ ∞ (π). The connection between partial differential operators and partial differential equations is given in this abstract setting as follows: For every h ∈ D k (π, π) and O ∈ Γ ∞ (X; π), the O-kernel ker O (h) of h is defined by ker O (h) := h −1 (im(O)) ⊂ J k (π), with the convention ker(h) := ker 0 (h), if π is a vector bundle. Observe that one has by definition The following fact is well-known:

It is clear that with an open subset
Next, we explain how the affine structure of π k−1,k can be used to introduce the notion of "operator symbols of (possibly nonlinear) partial differential operators". To avoid any confusion, we remark that with "symbol" we will exclusively mean "principal symbol" in this paper.
To this end, note that the assignment for a ∈ J k (π), v ∈ Sym k T * π k (a) X ⊗ker π |π 0,k (a) , is a (mono)morphism of vector bundles over J k (π). Note here that µ π k essentially extracts the pure k-th order part of vertical k-jets. Using the map µ π k , we can provide the following definition (see also [8]): Definition 3.11. For every h ∈ D k (π, π), the morphism σ(h) of vector bundles over h given by the composition Given a partial differential operator, one can use its symbol to check whether it defines a partial differential equation in the sense of Proposition 3.10 (see also Theorem 3.28 below): Proposition 3.12. Let h ∈ D k (π, π). If σ(h) is surjective, then so is h. If σ(h) is a submersion, then h is a submersion, too, which in particularly means that for every O ∈ Γ ∞ (X; π) with im(O) ⊂ im(h) the set ker O (h) ⊂ J k (π) is a partial differential equation.
Proof. We have the following commuting diagrams, where the maps for the second diagram are given by the tangential maps corresponding to the first one. If σ(h) = h V • µ π k is surjective, then so is h V and π V • h V , so that the first assertion follows from the first diagramm. If σ(h) is a submersion, then one can use the analogous argument for the second diagram to deduce that Th has full rank everywhere.

3.2.2.
Linear partial differential equations. We are now going to explain how the classical concepts of linear partial differential equations and partial differential operators fit into the general setting of Section 3.2.1. In fact, it will turn out that the notions "linear partial differential equation" and "linear partial differential operator" are equivalent under natural assumptions (this is only locally true in the nonlinear case [16]), and that the space of linear partial differential operators coincides with the space of classical linear partial differential operators (see Theorem 3.16 below). We begin with: Definition 3.13. Let π and π be vector bundles. a) A subset E ⊂ J k (π) is called a linear partial differential equation on π of order ≤ k, if π k|E : E → X is a sub-vector-bundle of π k . b) A morphism h : J k (π) → E of vector bundles over X is called a linear partial differential operator of order ≤ k from π to π. Remark 3.14. Let π and π be vector bundles. Then h ∈ D k (π, π) is linear, if and only if P h X : Γ ∞ (X; π) → Γ ∞ (X; π) is linear. We denote the linear space of linear partial differential operators by D k lin (π, π) ⊂ D k (π, π) and remark that if h ∈ D k lin (π, π), then one has h (l) ∈ D k+l lin (π, π l ) for all l ∈ N.

a) P • induces the isomorphism of linear spaces
lin (π, π) has constant rank, then ker(h) ⊂ J(π k ) is a linear partial differential equation. Conversely, if E ⊂ J k (π) is a linear partial differential equation, then there is a vector bundle π : E → X and an h ∈ D k lin π, π with constant rank such that E = ker h . Proof. a) We first have to show that P • lin is well-defined, which means that for any h ∈ D k lin (π, π), P h is in D k cl,lin (π, π). It is then clear that P • lin is a linear monomorphism. To this end, letx, e α , e β , ψ α be as in Definition 3.15 and let a α be a basis for F . Then we have the vector bundle chart so that we get the fibered chart of π as in Remark 3.2.3. Then (π k , u k ) : π −1 k (U) −→ U × F(m, 0, k) n is a vector bundle chart of π k by Lemma 3.6, and we get the frame e I,α ∈ Γ(U; π k ), α = 1, . . . n, I ∈ N m 0,k , given by e I,α := (π k , u k ) −1 (•, δ I,α ).
Hereby, δ I,α : N m 0,k → R n is defined by δ I,α (J) := 1 α , if I = J, and to be 0 elsewhere. Since h is a homomorphism of linear bundles over X, there are uniquely determined h α,β I ∈ C ∞ (U) such that one has for all α = 1, . . . , n, I ∈ N m 0,k and ψ α, The proof of the asserted well-definedness of P • lin is completed by observing that by the above construction of the frame e I,α for Γ ∞ (U; π k ) the following equality holds true: In order to prove surjectivity of P • lin , let D ∈ D k cl,lin (π, π), and let j k p ψ ∈ J(π k ), with p from an open subset U ⊂ X. Then h(j k p ψ) := D U ψ(p) gives rise to a well-defined element h ∈ D k lin (π, π), which of course satisfies P h lin = D. b) The first fact is well-known. For the second assertion, we can simply take E → X to be given by the quotient bundle J k (π)/E → X, and h to be given by the canonical projection J k (π) → J k (π)/E (see [25,Prop. 3.10] for a more general statement).
Finally, we explain in which sense the classical concept of linear operator symbols fits into the general setting of Section 3.2.1.
The following result recalls the classical definition of linear operator symbols and shows the naturality of Definition 3.11, in the sense that in the linear case, the linear operator symbol coincides with the operator symbol up to the canonical isomorphisms (3.15) and (3.16): Proposition and Definition 3.17. Let π and π be vector bundles and let h ∈ D k lin (π, π). a) There is a unique morphism of vector bundles over X with the following property: For every manifold chartx : U → R m of X for which there are frames e 1 , . . . , e n ∈ Γ ∞ (U; π) and e 1 , . . . , e n ∈ Γ ∞ (U; π), one has

18)
where v α I ∈ C ∞ (U), and where we have used the notation from Definition 3.15 and Theorem 3.16. The morphism σ lin (h) is called the linear operator symbol of h. b) The following diagram commutes, Here, one only has to prove that the representation (3.18) does not depend on a particular choice of local data. In fact, the easiest way to see this, is to note that one can simply define σ lin (h) by the diagram To see that σ lin (h) defined like this satsfies (3.18), letx : U → R m be a manifold chart of X such that there are frames e 1 , . . . , e n ∈ Γ ∞ (U; π), e 1 , . . . , e n ∈ Γ ∞ (U; π). Then, as in the proof of Theorem 3.16 a), picking a basis a α for F , we get the corresponding frame e I,α ∈ Γ ∞ (U; π k ), α = 1, . . . n, I ∈ N m 0,k , and we denote the representation of h with respect to e I,α and e β by h α,β I . Furthermore, by the proof of Theorem 3.16 a), we have h α,β I = (P h lin ) α,β I . Now one has which completes the proof of part a). b) Let a ∈ J k (π) be arbitrary, and letx : U → R m be a manifold chart of X around π k (a) such that there are frames e 1 , . . . , e n ∈ Γ ∞ (U; π), e 1 , . . . , e n ∈ Γ ∞ (U; π). Then, again as in the proof of Theorem 3.16 a), picking a basis a α for F and a basis a β for F , we get the corresponding adapted coordinates We can expand an arbitrary v ∈ Sym k T * π k (a) X ⊗ ker(Tπ |π 0,k (a) ) uniquely as and we arrive at Let us now evaluate σ π • σ(h) = σ π • h V • µ π k in v: By Proposition 3.5 and Lemma 3.6 we have the frame ∂ ∂u k I,α ∈ Γ ∞ π −1 k (U); (π k ) V , I ∈ N m 0,k , α = 1, . . . , n .
As h is a linear morphism, the linear morphism h V is represented with respect to the frames ∂ ∂u α k,I and ∂ ∂u β precisely by the functions where h α,β I ∈ C ∞ (U) is the representation of h with respect to (x, u) and (x, u) (cf. the proof of Theorem 3.16 a)). Thus, in view of , so that But this is equal to (3.20), in view of (P h lin ) α,β I = h α,β I . The claim follows.
3.3. The manifold of ∞-jets and formally integrable PDE's. Throughout Section 3.3, π will again be an arbitrary fiber bundle.
Finally, in this section we are going to make contact with the abstract theory on profinite dimensional manifolds from Section 2: We are going to prove that the space of "∞-jets" in π canonically becomes a profinite dimensional manifold (see Proposition 3.20), and that the space of "formal solutions" of a "formally integrable" partial differential equation on π canonically is a profinite dimensional submanifold of the latter (see Proposition 3.27).
We start by introducing the space of ∞-jets. In analogy to Definition 3.3, we have: Definition 3.18. Let p ∈ X. Any two ψ, ϕ ∈ Γ ∞ (p; π) are called ∞-equivalent at p, if ψ(p) = ϕ(p) and if for every fibered chart (x, u) : for all α = 1, . . . , n and all I ∈ N m with 1 ≤ |I| < ∞. The corresponding equivalence class j ∞ p ψ of ψ is called the ∞-jet of ψ at p. Remark 3. 19. In view of Remark 3.4, ∞-equivalence also only has to be checked in some fibered chart.
It will be convenient in what follows to set J −1 (π) := X, j −1 p ψ p := p, and π −1,0 := π. We define and obtain for every i ∈ Z ≥−1 a surjective map We equip J ∞ (π) with the initial topology with respect to the maps π i,∞ , i ∈ Z ≥−1 . Furthermore, we define C ∞ π to be the sheaf on J ∞ (π), whose section space C ∞ π (U) over an open U ⊂ J ∞ (π) is given by the set of all f ∈ C (U) such that for every In particular, (J ∞ (π), C ∞ π ) becomes a locally R-ringed space. Now observe that we have, in view of (3.7), a smooth projective system J i (π), π i,j , which graphically can be depicted by (3.23) together with a family of continuous maps These data have the following crucial property.
Proof. Let J ∞ (π) ′ := lim ←− i∈Z ≥−1 J i (π) denote the canonical projective limit of (3.23), that means let and let π ′ i,∞ : J ∞ (π) ′ → J i (π) denote the canonical projections. We are going to prove the existence of a homeomorphism Ξ such that the diagrams J ∞ (π) (3.24) commute for all i ∈ Z ≥−1 . Then the universal property of (J ∞ (π) ′ , π ′ i,∞ ) will directly imply the same property for (J ∞ (π), π i,∞ ), which is precisely (PFM1). As a consequence, (PFM2) is trivially satisfied by the definition of the structure sheaf C ∞ π . We now simply define Ξ(a) j := π j,∞ (a) for a ∈ J ∞ (π) and j ∈ Z ≥−1 . Then it is obvious that Ξ is a well-defined injective map, and that the Ξ-diagram in (3.24) commutes. In particular, the continuity of Ξ is directly implied by that of the maps π i,∞ . In order to see that Ξ is surjective and that Ξ −1 is continuous, let us recall that Borel's Theorem states that for any map there is a smooth functionψ : R m → R n such that t I = ∂ Iψ (0)/I! for all I ∈ N m . Let b ∈ J ∞ (π) ′ and letx : U → R m be a manifold chart of X around b −1 withx(b −1 ) = 0. Choosing furthermore a bundle chart φ : π −1 (U) → U × F and a manifold chartũ : B → R n of F , we get the fibered chart of π by Remark 3.2.3. Moreover, the function t : N m → R n , t I := u j,I (b j )/I!, if I ∈ N m 0,j , is well-defined. Borel's Theorem then produces a functionψ : R m → R n such that t I = ∂ Iψ (0)/I!. It is clear that the , and Ξ is surjective, indeed. Furthermore, by the construction of Ξ −1 (b), it is also clear that the Ξ −1 -diagram in (3.24) commutes, so that the continuity of Ξ −1 trivially follows from that of the π ′ i,∞ . This completes the proof. Next, we will prepare the introduction of formal integrability. Let us first note the following simple result: Proposition and Definition 3.21. a) For every h ∈ D k (π, π) there exists a unique h (l) ∈ D k+l (π, π l ) such that the following diagram of set theoretic sheaf morphisms commutes. The partial differential operator h (l) is called the l-jet prolongation of h. b) The partial differential operator Let E ⊂ J k (π) be an arbitrary partial differential equation for the moment. Since, by definition, the map π k|E : J k (π) ⊃ E −→ X is again a fibered manifold, there exists for every l ∈ N an obvious well-defined map ι l,E : J l (π k|E ) −→ J l (π k ), which comes from considering a locally defined section in π k|E as taking values in J k (π).
Definition 3.22. Let E ⊂ J k (π) be a partial differential equation. Then the set is called the l-jet prolongation of E.
If the underlying partial differential equation is actually given by a partial differential operator, then there is an explicit description of the corresponding l-jet prolongation ( [19], p. 294): Proposition 3.23. Let h ∈ D k (π, π) with constant rank and let O ∈ Γ ∞ (X; π) with im(O) ⊂ im(h). Then one has, for every l ∈ N, Let us note the simple fact that the following diagramm commutes, for every r ∈ N, Applying this in the case r = 1 implies π k+l,k+l+1 (E (l+1) ) ⊂ E (l) for any partial differential equation E ⊂ J k (π) and every l ∈ N, so that we obtain the maps Now we have the tools to give Definition 3.24. A partial differential equation E ⊂ J k (π) is called formally integrable, if E (l) is a submanifold of J k+l (π) and if (3.27) is a fibered manifold for every l ∈ N.
Remark 3.25. a) Here, it should be noted that E itself can always be considered as a trivial formally integrable partial differential equation on π of order 0, where in this case one has E (l) = J l (π) for all l ∈ N. b) Furthermore, there are abstract cohomological tests for partial differential equations to be formally integrable [19]. In fact, we will use such a test in the proof of Theorem 3.28 below; we refer the reader to [28] and particularly to [32] for the algorithmic aspects of these tests. Although it can become very involved to verify these test properties in particular examples, it is widely believed that most partial differential equations that arise naturally from geometry and physics are formally integrable. In accordance with the latter statement, Theorem 3.28 below states that all reasonable (possibly nonlinear) scalar partial differential equations are formally integrable. See for example [18] for a full treatement of the Yang-Mills-Higgs equations, and [23] for a treatement of Einstein's field equations under the viewpoint of formal integrability.
An important purely analytic consequence of formal integrability is given by the highly nontrivial Theorem 3.26 below, which essentially states that if all underlying data are real analytic, then formal integrability implies the existence of local analytic solutions with prescribed finite order Taylor expansions. Theorem 3.26 goes back to Goldschmidt [19] and heavily relies on (cohomological) results by Spencer [33] and Ehrenpreis-Guillemin-Sternberg [13]. This result can also be regarded as a variant of Michael Artin's Approximation Theorem [3].
Theorem 3.26. Assume that X is real analytic, that π is a real analytic fiber bundle (then so is π k ), and that E ⊂ J k (π) is formally integrable such that in fact π k,|E : E → X is a real analytic fibered submanifold of π k . Then, for every l ∈ N and a ∈ E (l) there exists an open neighborhood U ⊂ X of π k+l (a) and a real analytic solution ψ ∈ Γ ∞ (U; π) of E such that j k+l π k+l (a) ψ = a. Proof. This result follows directly from Theorem 9.1 in [19] (in combination with Proposition 7.1 therein).
Now let E ⊂ J k (π) be a formally integrable partial differential equation. Then we can define a subset E (∞) ⊂ J ∞ (π) by E (∞) := π −1 ∞,k (E). Inductively, one checks that the maps (3.22) restrict to surjective maps (3.29) so that In other words, this means that axioms (PFSM1) to (PFSM3) are satisfied for the subset E (∞) ⊂ J ∞ (π) and the smooth projective representation J i (π), π i,j , π i,∞ . Hence, one readily obtains Proposition and Definition 3.27. Let E ⊂ J k (π) be a formally integrable partial differential equation. Then J i (π), π i,j , π i,∞ induces on E (∞) the structure of a profinite dimensional submanifold of (J ∞ (π), C ∞ π ). In view of this fact, E (∞) will be called the manifold of formal solutions of E.
3.4. Scalar PDE's and interacting relativistic scalar fields. Let us first clarify that throughout Section 3.4, π : X × R → X will denote the canonical ine bundle.

3.4.1.
A criterion for formal integrability of scalar PDE's. We now come to the aforementioned result on formal integrability of scalar PDE's.
In the scalar situation, the sheaf of sections of π can be canonically identified with the sheaf of smooth functions on X. The smooth functions which are defined near p ∈ X will be denoted with C ∞ (p; X).
For any h ∈ D k (π, π), the space of vector bundle morphisms π * k Sym k (π T * X ) ⊗ π * k,0 V(π) −→ V(π k ) over h can be identified canonically as a linear space (remember here the maps (3.15) and (3.16)) with It follows that the symbol σ(h) of an h as above can be identified with is a linear map for every a ∈ J k (π). Thus, h induces the globally defined section σ(h) (1) of the vector bundle which, for every a ∈ J k (π), is given by Finally, we note that the space of k-th order partial differential operators D k (π, π) can be canonically identified as a linear space with C ∞ (J k (π)).
With these preparations, we have: Theorem 3.28. Let h ∈ C ∞ (J k (π)) and assume that the following assumptions are satisfied: (1) One has σ(h) |a = 0 for all a ∈ J k (π).
Then ker(h) ⊂ J k (π) is a formally integrable partial differential equation on π.
Remark 3.29. Note that assumption (1) together with Proposition 3.12 imply that ker(h) indeed is a partial differential equation, in particular, assumption (2) makes sense.
Proof of Theorem 3.28. The seemingly short proof that we are going to give actually combines two heavy machineries: The already mentioned abstract cohomological criterion for formal integrability of partial differential equations from [19], with a highly nontrivial reduction result for the cohomology of Cohen-Macaulay symbolic systems [23]. There seems to be no reasonable elementary proof of Theorem 3.28.
To prove our claim, let an arbitrary a ∈ J k (π) be given. By assumption (1), we can pick some v ∈ T * π k (a) X with σ(h) |a (v ⊗k ) = 0. Then, in the terminology of [23], π k (a) X C is a one-dimensional noncharacteristic subspace corresponding to the Cohen-Macaulay symbolic system g(h; a) given by ker(h) over a. Thus we may apply Theorem A from [23] to deduce that all Spencer cohomology groups H i,j (g(h; a)) except possibly H 0,0 (g(h; a)) and H 1,1 (g(h; a)) vanish. But now the result follows from combining (3) (2), the first prolongation becomes a vector bundle.
The assumptions (2) and (3) from Theorem 3.28 are technical regularity assumptions (which can become tedious to check in applications), whereas the reader should notice that assumption (1) therein is essentially trivial and means nothing but that the underlying differential operator globally is a "genuine" k-th order operator.

3.4.2.
Interacting relativistic scalar fields. As an application of Theorem 3.28, we will now consider evolution equations that correspond to (possibly nonlinearly!) interacting relativistic scalar fields on semiriemannian manifolds. To this end, let (X, g) be a smooth semiriemannian m-manifold with an arbitrary signature. The corresponding d'Alembert operator will be written as With functions F 1 , F 2 ∈ C ∞ (X), K ∈ C ∞ (R), we consider the partial differential operator h g,F 1 ,F 2 ,K ∈ C ∞ (J 2 (π)) given for p ∈ X, ϕ ∈ C ∞ (p; X) by h g,F 1 ,F 2 ,K j 2 p ϕ := ✷ g ϕ(p) + F 1 (p)ϕ(p) + F 2 (p)K(ϕ(p)). What we have in mind here is: Example 3.30. Let us assume that m = 4, that (X, g) has a Lorentz signature, and that F 1 = α 1 scal g + α 2 2 , F 2 = 0, K = α 3 K, where α 1 , α 3 ∈ R, α 2 ≥ 0, scal g ∈ C ∞ (X) denotes the scalar curvature of g and K ∈ C ∞ (R). Then ker(h g,F 1 ,0,K ) ⊂ J 2 (π) describes the onshell dynamics of a relativistic (real) scalar field with mass α 2 , where K is the field self-interaction with coupling strength α 3 , and where the number α 1 is an additional parameter, which is sometimes set equal to zero. For example, K(z) = z 3 corresponds to what is called a ϕ 4perturbation in the physics literature (since the corresponding potential in the Lagrange densitiy which has ker(h g,F 1 ,0,K ) as its Euler-Lagrange equation is given by V (ϕ) = ϕ 4 ). We refer the reader to [9] for the perturbative aspects of this equation in the flat ϕ 4 case.
Returning to the general situation, we can now prove the following result on scalar partial differential equations on semi-riemannian manifolds: Proposition 3.31. In the above situation, the assumptions (1), (2) and (3) from Theorem 3.28 are satisfied by h g,F 1 ,F 2 ,K . In particular, ker(h g,F 1 ,F 2 ,K ) ⊂ J 2 (π) is formally integrable, and the corresponding space of formal solutions canonically becomes a profinite dimensional manifold via Proposition 3.27. Moreover, if in addition (X, g), F 1 , F 2 and K are real analytic, then there exists for every l ∈ N and a ∈ ker(h g,F 1 ,F 2 ,K ) (l) an open neighborhood U ⊂ X of π k+l (a) and a real analytic solution ϕ ∈ C ∞ (U) of ker(h g,F 1 ,F 2 ,K ) such that j k+l π k+l (a) ϕ = a. Proof. In view of Theorem 3.26, we only have to prove that the assumptions (1), (2), (3) from 3.28 are satisfied. To this end, we set h := h g,F 1 ,F 2 ,K and assume F 2 = 0. Firstly, in view of , v ∈ T * π 2 (a) X, assumption (1) is obviously satisfied and ker(h) indeed is a partial differential equation. Analogously, to see that assumption (2) is satisfied, one just has to note that Thus, σ(h) (1) |a is surjective for fixed a. As a consequence of this and of being a vector bundle morphism, σ(h) (1) has constant rank.
Appendix A. Two results on completed projective tensor products Assume to be given two locally convex topological vector spaces V and W , and consider their algebraic tensor product V ⊗W . A topology τ on V ⊗ W is called compatible (in the sense of Grothendieck [20]) or a tensor product topology, if the following axioms hold true: (TPT1) V ⊗ W equipped with τ is a locally convex topological vector space which will be denoted by V ⊗ τ W . (TPT2) The canonical map V ×W → V ⊗ τ W is seperately continuous. (TPT3) For every equicontinuous subset A of the topological dual V ′ and every equicontinuous subset B of the topological dual W ′ , the set A ⊗ B := {λ ⊗ µ | λ ∈ A, µ ∈ B} is an equicontinuous subset of V ⊗ τ W ′ .
If τ is a tensor product topology on V ⊗ W , we denote by V ⊗ τ W the completion of V ⊗ τ W .
Example A.1. a) The projective tensor product topology is the finest locally convex vector space topology on V ⊗ W such that the canonical map V × W → V ⊗ W is continuous, cf. [20,35]. The projective tensor product topology is denoted by π. It is generated by seminorms p A ⊗ π q B , where p A , A ∈ A and q B , B ∈ B each run through a family of seminorms generating the locally convex topology on V respectively W , and p A ⊗ π q B is defined by The seminorm p A ⊗ π q B is in particular a cross seminorm, i.e. it satisfies the relation b) The injective tensor product topology on V ⊗ W , denoted by ε, is the locally convex topology inherited from the canonical embedding V ⊗ W ֒→ B s (V ′ s ⊗ W ′ s ), where B s (V ′ , W ′ ) denotes the space of seperately continuous bilinear forms on the product V ′ × W ′ of the weak topological duals V ′ and W ′ endowed with the topology of uniform convergence on products of equicontinuous subsets of V ′ and W ′ . See [20] and [35,Sec. 43] for details.
Remark A.2. a) By definition, the ε-topology on V ⊗ W is coarser than the π-topology. If V (or W ) is a nuclear locally convex topological vector space, then these two topologies coincide, cf. [20,35]. Since finite dimensional vector spaces over R are nuclear, this entails in particular that for finite dimensional V and W the natural vector space topology on V ⊗ W coincides with the (completed) π-and ε-topology. b) The projective tensor product, the injective tensor product, and their completed versions are in fact functors, so it is clear what is meant by f ⊗ ε g, f ⊗ π g, and so on, where f and g denote continuous linear maps.
Theorem A.3. Let V i i∈N and W i i∈N be two families of finite dimensional real vector spaces. Denote by V and W their respective product (within the category of locally convex topological vector spaces), i.e. let V := i∈N V i and W := i∈N W i .
Then V , W , and the completed projective tensor product V ⊗ π W are nuclear Fréchet spaces. Moreover, one has the canonical isomorphism Proof. Since each of the vector spaces V i and W i is a nuclear Fréchet space, and countable products of nuclear Fréchet are again nuclear Theorem A.4. Assume that V and W are projective limits of projective systems of finite dimensional real vector spaces V i , λ ij and W i , µ ij , respectively. Denote by the corresponding canonical maps. The completed π-tensor product V ⊗ π W together with the family of canonical maps then is a projective limit of the projective system V i ⊗ W i , λ ij ⊗ µ ij within the category of locally convex topological vector spaces. Moreover, both V and W are nuclear, hence V ⊗ π W = V ⊗ ε W .
Proof. First observe that V i ⊗ W i , λ ij ⊗ µ ij is a projective systems of finite dimensional real vector spaces, indeed. Next recall that projective limits of nuclear Fréchet spaces are nuclear by [35]. This proves the second claim. It remains to show the first one. To this end put V 0 := V 0 , W 0 := W 0 , and denote for every i ∈ N * by V i be the kernel of the map λ i−1i and by W i the kernel of µ i−1i . Morever, choose for every i ∈ N * a splitting f i : V i−1 → V i of λ i−1i , and a splitting g i : W i−1 → W i of µ i−1i . Put V := i∈N V i and W := i∈N W i .
Let π V i : V → V i be the projection onto the i-th factor of V , and π W j : W → W j the projection on the j-th factor of W . Now we inductively construct λ i : V → V i and µ i : W → W i . First, put λ 0 := π V 0 and µ 0 := π W 0 . Next, assume that we have constructed λ 0 , . . . , λ j and µ 0 , . . . , µ j such that for i ≤ k ≤ j (A.2) λ i = λ ik • λ k and µ i = µ ik • µ k .
Then ν i (z) ∈ V i for all i ∈ N, and ν : Z −→ V , z −→ ν i (z) i∈N is well-defined, linear, and continuous. Moreover, it follows by induction on i ∈ N that λ i ν = ν i . For i = 0 this is clear, so assume that we have shown this for some i ∈ N. Then, for z ∈ Z, which finishes the inductive argument. Assume that ν ′ : Z → V is another continuous linear map such that λ i ν ′ = ν i for all i ∈ N. First, this entails that π V 0 ν ′ = λ 0 ν ′ = ν 0 = ν 0 . Assume that π V i ν ′ = ν i for some i ∈ N. Then π V i+1 ν ′ = λ i+1 ν ′ − f i+1 λ i ν ′ = ν i+1 − f i+1 ν i = ν i+1 . Hence, one obtains, for all i ∈ N, π V i ν ′ = ν i = π V i ν, which proves ν ′ = ν. So V is a projective limit of V i , λ ij , and W a projective limit of W i , µ ij . Moreover, V is canonically isomorphic to V , and W to W .
The theorem is now proved, if we can show that V ⊗ π W together with the family of canonical maps λ i ⊗ π µ i : V ⊗ π W → V i ⊗ W i is a projective limit of the projective system V i ⊗ W i , λ ij ⊗ µ ij . But this is clear, since by the preceeding theorem, and, for k ∈ N,