Pfaffian Fibrations: Intrinsic PDE Geometry

• Pfaffian fibrations are intrinsic geometric structures that encode PDEs via a Cartan-type 1-form or a transverse distribution, eliminating the need for an ambient jet bundle.
• They offer dual presentations—in form language and distributional language—that facilitate precise analysis of prolongations, integrability, and symmetries in differential equations.
• Their framework underpins linearisation methods, relative algebroid constructions, and connections to Lie pseudogroup symmetries, thus enhancing the study of solution spaces.

Pfaffian fibrations are an intrinsic geometric framework for encoding the essential data of a partial differential equation without privileging an ambient jet bundle. In the formulation developed in “From PDEs to Pfaffian fibrations” (Cattafi et al., 2019), the basic object is a fibration equipped with a Cartan-type $1$-form, or equivalently a distinguished distribution whose holonomic sections play the role of solutions. Subsequent work relating Pfaffian fibrations to relative algebroids shows that they also admit a canonical infinitesimal model with the same prolongations and local solutions, and that this model is compatible with natural notions of symmetry (Smilde, 1 Oct 2025).

1. Definition and equivalent formulations

A Pfaffian fibration may be presented in either form language or distributional language. In the form-theoretic presentation, a Pfaffian fibration $(P,\theta)$ over a manifold $M$ consists of a fibration $\pi:P\to M$ together with a pointwise surjective differential form

$\theta\in\Omega^1(P,N),$

where $N\to P$ is a vector bundle, such that $\theta$ is $\pi$-regular and $\pi$-involutive (Cattafi et al., 2019). The $\pi$-regularity condition is

$(P,\theta)$0

so the kernel distribution is transverse to the $(P,\theta)$1-fibres, while $(P,\theta)$2-involutivity requires the symbol distribution

$(P,\theta)$3

to be involutive in the Frobenius sense (Cattafi et al., 2019).

The equivalent distributional presentation, emphasized in the later symmetry-theoretic treatment, describes a Pfaffian fibration as a triple

$(P,\theta)$4

where $(P,\theta)$5 is a submersion and $(P,\theta)$6 is a distribution satisfying the transversality condition

$(P,\theta)$7

and the $(P,\theta)$8-involutivity condition

$(P,\theta)$9

is involutive (Smilde, 1 Oct 2025). The passage between the two descriptions is immediate: if $M$0, then $M$1 is a $M$2-transversal distribution with involutive symbol, and conversely a distribution $M$3 determines a surjective quotient form

$M$4

(Cattafi et al., 2019).

This dual description is structurally important. The form viewpoint makes contact with Cartan forms and Spencer-operator-type constructions, whereas the distribution viewpoint is adapted to prolongations, relative algebroids, and symmetry groupoids. A local section is holonomic precisely when it annihilates the Pfaffian form or, equivalently, when its tangent image lies in the Pfaffian distribution: $M$5 (Cattafi et al., 2019); equivalently, for $M$6,

$M$7

(Smilde, 1 Oct 2025).

2. Relation to partial differential equations and jet bundles

The motivating example is the Cartan geometry of jet bundles. For a submersion $M$8, the first jet bundle $M$9 with its Cartan distribution $\pi:P\to M$0 is a Pfaffian fibration

$\pi:P\to M$1

and any PDE $\pi:P\to M$2 becomes a Pfaffian fibration by restriction: $\pi:P\to M$3 (Smilde, 1 Oct 2025). In the general $\pi:P\to M$4-th order setting, if $\pi:P\to M$5 is a PDE on a fibration $\pi:P\to M$6, then the restricted Cartan form $\pi:P\to M$7 already captures the essential geometric structure, so the pair

$\pi:P\to M$8

is the intrinsic object of interest (Cattafi et al., 2019).

The central claim of the intrinsic theory is that the ambient jet bundle is not the essential datum. A section $\pi:P\to M$9 of $\theta\in\Omega^1(P,N),$0 solves the PDE exactly when its jet prolongation is tangent to $\theta\in\Omega^1(P,N),$1, equivalently

$\theta\in\Omega^1(P,N),$2

or, in Pfaffian terms,

$\theta\in\Omega^1(P,N),$3

(Cattafi et al., 2019). This is the precise sense in which a PDE can be encoded by a fibration together with a Cartan-type $\theta\in\Omega^1(P,N),$4-form.

The standard slogan recorded in the foundational paper is that a PDE is encoded by a fibration together with a geometric device that detects solutions, and the jet bundle is background structure rather than essential structure (Cattafi et al., 2019). This suggests that Pfaffian fibrations isolate precisely those aspects of PDE geometry needed for prolongation, integrability, and linearisation, while discarding extraneous choices of embedding.

3. Holonomic sections, curvature, prolongations, and integrability

The intrinsic solution concept is given by holonomic sections. A section $\theta\in\Omega^1(P,N),$5 is holonomic if

$\theta\in\Omega^1(P,N),$6

and a Pfaffian fibration is PDE-integrable if through every point $\theta\in\Omega^1(P,N),$7 there exists a local holonomic section passing through $\theta\in\Omega^1(P,N),$8 (Cattafi et al., 2019). In the distributional formulation, holonomicity is the condition that the tangent map of the section lands in the Pfaffian distribution (Smilde, 1 Oct 2025).

Curvature controls the failure of the Pfaffian distribution to be integrable. For a Pfaffian distribution $\theta\in\Omega^1(P,N),$9, the curvature is

$N\to P$0

or, in form language,

$N\to P$1

(Cattafi et al., 2019). In the later notation, the curvature map entering prolongation theory is

$N\to P$2

(Smilde, 1 Oct 2025).

Two first-order prolongation spaces are distinguished. The partial prolongation is

$N\to P$3

while the full prolongation is

$N\to P$4

(Smilde, 1 Oct 2025). In the earlier notation, the partial prolongation $N\to P$5 is the space of partial integral elements,

$N\to P$6

and the classical prolongation is the space of integral elements,

$N\to P$7

(Cattafi et al., 2019).

These prolongation spaces are not merely formal definitions. The paper proves that $N\to P$8 is smooth and that the projection

$N\to P$9

is an affine bundle modeled on

$\theta$0

(Cattafi et al., 2019). Moreover, $\theta$1 is a smooth affine subbundle of $\theta$2 if and only if the torsion vanishes and the first prolongation of the symbol bundle has constant rank; in that case its model vector bundle is $\theta$3, and the projection to $\theta$4 is the largest normalised prolongation (Cattafi et al., 2019).

Formal integrability is defined by iterating prolongation. In the later terminology, a Pfaffian fibration is $\theta$5-integrable if $\theta$6 is a submersion, $\theta$7-integrable if all prolongations up to order $\theta$8 exist, and formally integrable if it is $\theta$9-integrable for all $\pi$0 (Smilde, 1 Oct 2025). The foundational paper gives a recursive obstruction theory in terms of higher torsions and symbol prolongations, and extends Goldschmidt’s criterion: if the symbol space $\pi$1 is $\pi$2-acyclic,

$\pi$3

and $\pi$4 is smooth while the first prolongation is surjective, then the Pfaffian fibration is formally integrable (Cattafi et al., 2019).

4. Linear theory, relative connections, and linearisation

The linear theory of Pfaffian fibrations is modeled on linear PDEs and Spencer operators. If $\pi$5 is a vector bundle, a linear Pfaffian fibration may be described either by a pointwise surjective linear form

$\pi$6

or by a linear distribution $\pi$7; there is a $\pi$8-to-$\pi$9 correspondence between the two descriptions (Cattafi et al., 2019).

The form-theoretic description is equivalent to a relative connection

$\pi$0

with symbol map $\pi$1, satisfying the Leibniz rule

$\pi$2

(Cattafi et al., 2019). In the classical jet-bundle case, this recovers the Spencer operator, so linear Pfaffian fibrations supply the intrinsic language behind standard linear PDE operators.

Linearisation along a holonomic section provides the nonlinear-to-linear passage. Given a Pfaffian fibration $\pi$3 and a holonomic section $\pi$4, the linearisation is

$\pi$5

where $\pi$6 is a relative connection obtained by differentiating families of sections through $\pi$7 (Cattafi et al., 2019). For a linear Pfaffian fibration, linearisation at the zero section recovers the original linear object, and for jet bundles the construction recovers the classical Spencer operator (Cattafi et al., 2019).

This identifies the linearised Pfaffian geometry of a solution with the familiar infinitesimal theory of PDEs. A plausible implication is that the Pfaffian-fibration formalism is designed not only to encode solution spaces intrinsically, but also to preserve the standard deformation-theoretic structures used in the analysis of formal integrability.

5. Relative algebroids and the formal equivalence theorem

A principal development in the later literature is the construction of a canonical relative algebroid underlying any Pfaffian fibration. Starting from a Pfaffian fibration $\pi$8, the relevant geometric data are the distribution $\pi$9 and the foliation

$\pi$0

(Smilde, 1 Oct 2025). The associated vector bundle is

$\pi$1

which carries a canonical flat $\pi$2-connection, and from the geometry of $\pi$3 one obtains a relative derivation

$\pi$4

(Smilde, 1 Oct 2025). The construction is described concretely by

$\pi$5

for flat forms $\pi$6, where $\pi$7 and $\pi$8 are the canonical maps induced by the splitting of the normal bundle (Smilde, 1 Oct 2025). The resulting statement is that a Pfaffian fibration $\pi$9 gives rise to a relative algebroid

$(P,\theta)$00

over the foliation $(P,\theta)$01 (Smilde, 1 Oct 2025).

The main structural result is that Pfaffian fibrations and their underlying relative algebroids have the same formal theory. The theorem states that the partial prolongation spaces of $(P,\theta)$02 and $(P,\theta)$03 are canonically isomorphic, hence the Pfaffian fibration is $(P,\theta)$04-integrable if and only if the relative algebroid is; that their tableau maps are canonically identified; that if $(P,\theta)$05 is $(P,\theta)$06-integrable then the relative algebroid underlying $(P,\theta)$07 is canonically isomorphic to the prolongation of the original relative algebroid; and that germs of holonomic sections are in one-to-one correspondence with germs of realizations of the relative algebroid, modulo diffeomorphism (Smilde, 1 Oct 2025).

The theorem’s content is explicit: the spaces of first-order admissible jets coincide, the symbol or tableau data coincide, the prolongation towers coincide when defined, and the local solutions coincide up to the expected equivalence on the algebroid side (Smilde, 1 Oct 2025). This shows that the relative algebroid is not merely an associated invariant. It captures the same formal PDE data as the original Pfaffian fibration.

Naturality is also built into the construction. A morphism of Pfaffian fibrations

$(P,\theta)$08

satisfying

$(P,\theta)$09

induces a bundle map

$(P,\theta)$10

which is a morphism of relative algebroids (Smilde, 1 Oct 2025). This establishes a functorial passage from Pfaffian fibrations to relative algebroids.

6. Symmetries, Pfaffian groupoids, and terminological scope

The symmetry theory distinguishes two notions. Internal symmetries are local diffeomorphisms preserving the Cartan distribution,

$(P,\theta)$11

whereas Pfaffian symmetries are local diffeomorphisms preserving both $(P,\theta)$12 and its vertical part,

$(P,\theta)$13

(Smilde, 1 Oct 2025). The distinction is essential because a Pfaffian symmetry need not preserve $(P,\theta)$14, so it need not be a morphism of Pfaffian fibrations over the base (Smilde, 1 Oct 2025).

Prolongation links the two notions. An internal symmetry $(P,\theta)$15 prolongs to a local map

$(P,\theta)$16

and, if $(P,\theta)$17 is smooth, to

$(P,\theta)$18

(Smilde, 1 Oct 2025). The main theorem on this point states that if $(P,\theta)$19, then $(P,\theta)$20 is a Pfaffian symmetry of $(P,\theta)$21; if $(P,\theta)$22 is $(P,\theta)$23-integrable, then $(P,\theta)$24 is also a Pfaffian symmetry of $(P,\theta)$25; any Pfaffian symmetry of $(P,\theta)$26 is locally the prolongation of an internal symmetry of $(P,\theta)$27; and if $(P,\theta)$28 is $(P,\theta)$29-integrable and the tableau map is involutive, then every Pfaffian symmetry of $(P,\theta)$30 is locally the prolongation of an internal symmetry of $(P,\theta)$31 (Smilde, 1 Oct 2025).

The symmetry pseudogroups are organized into Pfaffian groupoids. The internal symmetry groupoid is

$(P,\theta)$32

and the Pfaffian symmetry groupoid is

$(P,\theta)$33

(Smilde, 1 Oct 2025). These inherit the Cartan distribution $(P,\theta)$34 from $(P,\theta)$35, and the canonical representation

$(P,\theta)$36

leaves the derivation $(P,\theta)$37 invariant (Smilde, 1 Oct 2025). In particular, Pfaffian symmetries induce local isomorphisms of the underlying relative algebroid and preserve both the flat connection and the derivation (Smilde, 1 Oct 2025).

The framework extends to actions of arbitrary Pfaffian groupoids $(P,\theta)$38 on a Pfaffian fibration $(P,\theta)$39 with moment map $(P,\theta)$40. An action is by internal symmetries if

$(P,\theta)$41

and by Pfaffian symmetries if, in addition,

$(P,\theta)$42

(Smilde, 1 Oct 2025). If $(P,\theta)$43 is an action by Pfaffian symmetries and $(P,\theta)$44 is $(P,\theta)$45-integrable, then the induced action

$(P,\theta)$46

is by symmetries of relative algebroids (Smilde, 1 Oct 2025). The paper identifies applications in particular to partial differential equations with Lie pseudogroup symmetries (Smilde, 1 Oct 2025).

A recurrent source of confusion is terminological. Several other works use the word “Pfaffian” in unrelated senses: for example, one paper explicitly states that it does not use the phrase “Pfaffian fibration” in the differential-geometric sense, even though it studies a fibration of odd orthogonal Grassmannians whose degeneracy classes are governed by Pfaffian formulas (Hudson et al., 2016); another emphasizes that its “Pfaffian structure” concerns random-matrix partition functions rather than a geometric fibration in the differential-geometric sense (Babinet et al., 2022). In the PDE literature, by contrast, a Pfaffian fibration means precisely the intrinsic geometric object defined by a fibration together with a Pfaffian distribution or form satisfying transversality and involutivity conditions (Cattafi et al., 2019).

The resulting picture is that Pfaffian fibrations form a bridge between nonlinear PDEs, prolongation theory, Spencer-type linearisation, relative algebroids, and Lie pseudogroup symmetry. This suggests that their role is not merely terminological but foundational: they provide the intrinsic object on which formal integrability, infinitesimal structure, and symmetry reduction can all be formulated in a common language.