Double Fibration Transform

• Double fibration transform is a geometric and analytic construct that integrates functions over families of submanifolds via two surjective submersions, generalizing classical Radon and X-ray transforms.
• It employs a precisely defined incidence manifold and the Bolker condition to ensure microlocal invertibility and stable recovery of analytic singularities.
• Applications include inverse boundary value problems, analytic tomography, and holographic reconstruction, bridging integral geometry and mathematical physics.

A double fibration transform is a geometric and analytic construction that arises by integrating functions (or sections) over families of submanifolds parameterized by a suitable double fibration diagram. It generalizes the classical Radon and X-ray transforms and possesses a rich microlocal and algebraic structure. In its most general analytic setting, a double fibration transform is defined for submanifolds $Z \subset G \times X$ with two surjective submersions onto $G$ and $X$, enabling the integral transformation of functions or distributions on $X$ to functions on $G$, frequently governed by an analytic or geometric weight. Double fibration transforms have deep applications in integral geometry, inverse problems, representation theory, and mathematical physics.

1. Geometric and Analytic Structure of Double Fibration Transforms

The foundational setting for a double fibration transform requires oriented smooth manifolds $G$ and $X$ (dimensions $N, n$), and an embedded submanifold $Z \subset G \times X$ with $\dim Z = N + n'$ ($G$0), such that

• Both projections $G$1 and $G$2 are surjective submersions.
• For each $G$3, the fiber $G$4 is a smooth $G$5-dimensional submanifold of $G$6.
• For each $G$7, the fiber $G$8 is a smooth $G$9-dimensional submanifold of $X$0, where $X$1.

Given a real-analytic, nowhere vanishing weight $X$2 and appropriate volume forms, the analytic double fibration transform is

$X$3

mapping compactly supported analytic functions (or distributions) on $X$4 into analytic functions on $X$5 (Mazzucchelli et al., 2023, Chihara, 2024).

In the smooth category, the Schwartz kernel of $X$6 is a conormal distribution associated to $X$7, and $X$8 is a Fourier integral operator (FIO) of appropriate order, with canonical relation determined by the geometry of $X$9.

2. Microlocal and Ellipticity Conditions: The Bolker Condition

Central to the invertibility and stability theory of double fibration transforms is the Bolker condition (in the sense of Guillemin) (Mazzucchelli et al., 2023). In microlocal terms, the canonical relation $X$0 governs the propagation of singularities. The Bolker condition at $X$1 requires:

• $X$2 is locally injective,
• $X$3 has maximal rank (is an immersion) at the point.

This ensures that $X$4 is an elliptic FIO: the transform is microlocally invertible, and analytic singularities of $X$5 away from the so-called $X$6-conjugate locus can be detected and reconstructed from $X$7. This framework retains deep analogies with the theory of the Radon and X-ray transforms but accommodates general submanifold and weight structures (Chihara, 2024, Chihara et al., 30 Jun 2025).

3. Invertibility, Singularities, and Support Theorems

Under the Bolker condition, analytic double fibration transforms admit strong support and inversion results. If $X$8 is analytic and elliptic, then for any distribution $X$9, absence of analytic singularities of $G$0 at $G$1 implies absence of singularities of $G$2 at $G$3: $G$4 (Mazzucchelli et al., 2023, Chihara et al., 30 Jun 2025). A Helgason-type support theorem follows: vanishing of $G$5 in open sets propagates analyticity and can induce global vanishing by unique continuation. In higher regularity, injectivity and stability can be established for real-analytic data, with stability up to Hölder modulus (Busch et al., 27 Oct 2025).

When the Bolker condition fails, e.g., in the presence of conjugate (focal) points, more intricate microlocal phenomena arise. The normal operator $G$6 decomposes into an elliptic pseudodifferential part and a finite sum of FIOs, each associated to the loci of conjugate points of various degrees, and inversion is obstructed for certain singularities (Chihara, 2024).

4. Paradigmatic Examples and Generalizations

• Classical Radon and X-ray Transforms: The Radon transform integrates over $G$7-planes in $G$8, and the X-ray transform is the $G$9 case. Here, the double fibration diagram is manifestly geometric, and the Bolker condition holds globally, yielding an elliptic normal operator and explicit inversion formulas (Hilgert et al., 2013, Chihara, 2024).
• Minimal Surface Transform: The linearized forward operator for the boundary rigidity problem is a minimal surface transform, interpreted as a double fibration transform over the space of embedded minimal surfaces in a compact manifold with boundary. Under analytic or foliation (convexity) conditions, injectivity and determination of the analytic wave front set are obtained, providing new approaches to rigidity and holographic reconstruction problems (AdS/CFT) (Busch et al., 27 Oct 2025).
• Matrix-weighted and Nonabelian Ray Transforms: Generalizing to vector-valued functions with matrix weights, the matrix-weighted double fibration transform encompasses nonabelian transport operators. On analytic surfaces with strictly convex boundary, the matrix-weighted X-ray transform is injective and uniquely determines, for example, real-analytic Higgs fields from their parallel transport data (Chihara et al., 30 Jun 2025).
• Desingularized and b-Fibrations: For manifolds with boundary, double b-fibrations provide a resolution of singularities for the X-ray/Radon transforms near the boundary and enable sharp mapping theorems for polyhomogeneous conormal spaces and normal operator analysis (Mazzeo et al., 2021, Hansen, 11 Mar 2026).

5. Algebraic and Cohomological Variants: Flag Domains and Penrose-type Transforms

In complex geometry, especially for flag domains and cycle spaces, the double fibration transform appears as a cohomological (not integral) correspondence (Eastwood, 2012, Eastwood et al., 2012). Here, the transform relates Dolbeault cohomology on a parameter space $G$0 to global sections on a manifold $G$1 via a double fibration $G$2: $G$3 The transform is governed by a spectral sequence, with Serre-type duality and explicit criteria for injectivity and range in terms of representation theoretic data. The Penrose transform (and its duals) in twistor theory appear as special cases, yielding concrete analytic realization of infinite-dimensional representations and invariant differential complexes on symmetric spaces (Eastwood et al., 2012).

6. Applications in Geometry, Mathematical Physics, and Inverse Problems

Double fibration transforms underpin key results in:

• Inverse boundary value and rigidity problems: Determination of Riemannian metrics from boundary measurements, via the minimal surface transform (Busch et al., 27 Oct 2025).
• Analytic tomography and support theorems: Unique recovery of analytic singularities and support from data, with extensions to ray transforms for real principal-type PDEs and wave propagation (Mazzucchelli et al., 2023).
• AdS/CFT bulk reconstruction: The minimal surface transform provides the geometric underpinning for holographic entanglement entropy calculations (Ryu–Takayanagi prescription), relating bulk fields to boundary CFT observables (Busch et al., 27 Oct 2025).
• Representation theory and complex geometry: Realization of cohomological data and unitary representations via double fibration transforms in the algebraic and complex analytic categories (Eastwood et al., 2012, Eastwood, 2012).

7. Structural Summary and Theoretical Outlook

Double fibration transforms unify a broad class of integral, analytic, and cohomological correspondences: from concrete geometric integral transforms (Radon, X-ray, minimal surfaces), to complex analytic spectral sequences, to applications in inverse problems and quantum field theory. The analytic theory is fundamentally governed by the geometry of the incidence manifold $G$4, microlocal properties encoded in the Bolker condition, and singularity propagation via FIO calculus. Open directions involve further generalizations to nonanalytic settings, refined microlocal decomposition in the presence of degenerate geometry, and connections to modern problems in mathematical physics and representation theory.

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References:

• "A general support theorem for analytic double fibration transforms" (Mazzucchelli et al., 2023)
• "Microlocal analysis of double fibration transforms with conjugate points" (Chihara, 2024)
• "The matrix weighted real-analytic double fibration transforms" (Chihara et al., 30 Jun 2025)
• "Generalized boundary rigidity and minimal surface transform" (Busch et al., 27 Oct 2025)
• "Double b-fibrations and desingularization of the X-ray transform on manifolds with strictly convex boundary" (Mazzeo et al., 2021)
• "Polyhomogeneous mapping properties of the Radon transform and backprojection operator on the unit ball" (Hansen, 11 Mar 2026)
• "A double fibration transform for complex projective space" (Eastwood, 2012)
• "A duality for the double fibration transform" (Eastwood et al., 2012)
• "The Radon transform and its dual for limits of symmetric spaces" (Hilgert et al., 2013)