Double Fibration Transforms

Updated 17 December 2025

Double fibration transforms are integral operators that map functions via incidence structures, unifying diverse transforms like Radon and X-ray.
They rely on analytic Fourier integral operator theory and the Bolker condition to guarantee injectivity and accurate microlocal inversion.
Applications span geodesic, twistor, and data-driven recovery problems, demonstrating robust methods for capturing analytic wavefront sets and handling conjugate points.

A double fibration transform is an integral operator defined on the geometric data of a double fibration—an incidence structure involving two manifolds and an intermediate submanifold whose projections behave as submersions. Such transforms encompass and unify a wide spectrum of geometric integral operators across real, complex, and analytic categories, including Radon transforms, geodesic X-ray transforms, twistor correspondences, and their matrix-weighted or nonabelian extensions. They are characterized as analytic Fourier integral operators (FIOs) associated to the conormal bundle of the incidence submanifold, with fundamental implications for injectivity, microlocal analysis, range characterization, and modern learning theory.

1. Geometric Structure of Double Fibrations and Transform Definition

A double fibration consists of manifolds $G$ and $X$ (with $\dim G = m$ , $\dim X = n$ , $m \geq n$ ), and an embedded real-analytic (or holomorphic) submanifold $Z \subset G \times X$ of intermediate dimension. The two coordinate projections, $\pi_G: Z \to G$ and $\pi_X: Z \to X$ , are required to be submersions:

$\begin{array}{ccc} & Z & \ \pi_G \swarrow & & \searrow \pi_X \ G & & X \end{array}$

For each $z \in G$ , the fiber $G_z := \pi_X(\pi_G^{-1}(z)) \subset X$ is an $n'$ -dimensional submanifold, serving as the locus of integration. The (possibly matrix-weighted) double fibration transform for compactly supported smooth (or vector-valued) functions is

$R_W f(z) = \int_{G_z} W(z, x) f(x) d\omega_{G_z}(x), \quad z \in G,$

where $W$ is a weight taking values in $GL(N, \mathbb{C})$ and $\omega_{G_z}$ is the real-analytic orientation form on $G_z$ (Chihara et al., 30 Jun 2025).

In complex geometry, similar structures appear with holomorphic submersions and cycles, forming the backbone of the Penrose/twistor transform and associated spectral-sequence constructions (Eastwood et al., 2012, Eastwood, 2012).

2. Microlocal Structure and the Bolker Condition

The Schwartz kernel of a double fibration transform is an analytic Lagrangian (or conormal) distribution associated to $N^*Z$ in $T^*G \times T^*X$ . The transform is a FIO with canonical relation defined by the twisted conormal bundle

$C := \{ (z, \zeta, x, \eta) \mid (z, \zeta, x, -\eta) \in N^*Z \}.$

Central to microlocal invertibility is the Bolker condition, which requires that the left projection $\pi_L: C \to T^*G$ is a local injective immersion—a necessary and sufficient condition for the possibility of recovering analytic singularities (Mazzucchelli et al., 2023). In the case of curve or ray transforms, this condition reduces to the absence of conjugate points and intersection properties of associated submanifolds (Chihara, 19 Dec 2024). Failure of the Bolker condition leads to the presence of conjugate loci and a more intricate structure in the normal operator, including additional non-elliptic FIO terms.

3. Analytic Wavefront Set Determination and Support Theorems

Analytic double fibration transforms allow the analytic wavefront set $\mathrm{WF}_a(f)$ of $f$ to be controlled via the transform:

$(z, \zeta) \notin \mathrm{WF}_a(Rf) \implies (x, \eta) \notin \mathrm{WF}_a(f),$

provided the Bolker condition holds at $(z, \zeta, x, \eta) \in C$ (Chihara et al., 30 Jun 2025, Mazzucchelli et al., 2023).

This propagation result is established using analytic wave packet/FBI transform methods, stationary phase in the fiber variables, and analytic microlocal techniques of Hörmander and Sjöstrand. As a consequence, local vanishing of $Rf$ on open sets implies vanishing of $f$ on associated regions, yielding analytic support and uniqueness theorems. Layer-stripping arguments promote local results to global injectivity on manifolds foliated by strictly convex analytic hypersurfaces (Chihara et al., 30 Jun 2025, Mazzucchelli et al., 2023).

4. Matrix-Weighted, Nonabelian, and Holomorphic Extensions

Matrix weights and nonabelian structures appear naturally in physical and geometric inverse problems. For instance, in the nonabelian X-ray transform, given real-analytic Higgs fields $\Phi, \Psi \in C^\omega(M; \mathbb{C}^{N \times N})$ on a real-analytic Riemannian manifold $M$ with strictly convex boundary point, one studies:

$C_\Phi(z) := U(z, 0),$

where $U(z, t)$ solves a transport ODE with coefficients $\Phi(x_z(t))$ . Pseudo-linearization shows that equality of scattering data for $\Phi$ and $\Psi$ implies the vanishing of a matrix-weighted transform, and analytic propagation yields uniqueness of $\Phi$ (Chihara et al., 30 Jun 2025).

In holomorphic settings, double fibration transforms relate to Dolbeault cohomology classes via pullback and restriction to fibers, with the transform realized as pushforward (integral) over cycles. The governing spectral sequence collapses under vanishing conditions for higher direct images, yielding injectivity or explicit differential operator characterizations of the range (Eastwood et al., 2012, Eastwood, 2012).

5. Normal Operators, Conjugate Points, and Microlocal Decomposition

The normal operator $N = R^*R$ combines the structure of the transform and its adjoint; in the absence of conjugate points and with the Bolker condition, $N$ is elliptic and invertible (typically a pseudodifferential operator of negative order, e.g., $N \in \Psi^{-1}$ for X-ray transform on surfaces) (Chihara, 19 Dec 2024, Mazzeo et al., 2021).

In the presence of conjugate points (i.e., failure of the Bolker condition at certain loci), $N$ splits as

$N = P + \sum_{k=1}^n \sum_\alpha F_{k, \alpha},$

where $P$ is elliptic and each $F_{k, \alpha}$ is a FIO of order $-(n + n - k)/2$ associated with conjugate loci of degree $k$ (Chihara, 19 Dec 2024). The structure and regularity of these components, and their influence on the mapping properties and inversion, depend on the stable regularity of the conjugate strata. In boundary settings, Melrose's double $b$ -fibration and pushforward theorems provide explicit index set transformations and smoothness criteria for normal and adjoint operators (Mazzeo et al., 2021).

6. Extensions: Learning Theory and Data-Driven Recovery

Recent advances apply the double fibration formalism to scientific machine learning, treating integral operators (of double fibration type) as the learning targets. If the Bolker condition holds, double fibration transforms are learnable from input-output data with superalgebraic rates (i.e., not subject to the curse of dimensionality). Neural architectures encoding the geometric structure—specifically, via level-set parametrizations and cross-attention-like continuous kernels—are universal, stable, and interpretable (Roddenberry et al., 10 Dec 2025).

For $R$ a double fibration transform on compact manifolds $X$ , $Y$ , to achieve error $\epsilon$ requires only $O(\epsilon^{-1/r} \log(1/\epsilon))$ samples for any $r > 1$ . Deep networks using factorizations of the incidence function efficiently approximate $R$ , with provable error bounds and robustness to discretization.

7. Applications and Explicit Examples

The double fibration paradigm encompasses:

Geodesic and matrix-weighted X-ray transforms on strictly convex Riemannian manifolds (Chihara et al., 30 Jun 2025, Mazzeo et al., 2021)
Generalized Radon and $d$ -plane transforms in Euclidean space (Chihara, 19 Dec 2024, Mazzucchelli et al., 2023)
Null bicharacteristic (light-ray) transforms for real principal type PDEs (Mazzucchelli et al., 2023)
Twistor Penrose transforms mapping Dolbeault cohomology to differential invariant solution spaces (Eastwood, 2012, Eastwood et al., 2012)
Learning and recovery of unknown geometric operators from data, with precise sample complexity and universal architectural frameworks (Roddenberry et al., 10 Dec 2025)

These diverse instances are unified by the incidence geometry of $Z \subset G \times X$ , analytic FIO structure, the Bolker condition, and the propagation of singularities governed by the canonical relation.