X-Ray Transform in Asymptotically Conic Spaces

Abstract: In this article, we study the properties of the geodesic X-ray transform for asymptotically Euclidean or conic Riemannian metrics and show injectivity under non-trapping and no conjugate point assumptions. We also define a notion of lens data for such metrics and study the associated inverse problem.

• The paper proves injectivity of the geodesic X-ray transform for functions and symmetric tensors on asymptotically conic manifolds.
• It employs microlocal methods, including gauge reduction and Pestov identities, to derive sharp stability estimates and rigidity results.
• The study extends classical inverse problems to non-compact settings, establishing unique metric recovery from scattering data under non-trapping conditions.

X-Ray Transform in Asymptotically Conic Spaces: Injectivity and Rigidity

Abstract and Context

The paper "X-Ray Transform in Asymptotically Conic Spaces" (1910.09631) conducts a rigorous analysis of the geodesic X-ray transform on asymptotically Euclidean and, more generally, asymptotically conic Riemannian manifolds. It establishes injectivity under mild global geometric assumptions, introduces a generalized notion of lens data for these non-compact settings, and derives rigidity results for the associated inverse problems. The established results extend classical rigidity and integral geometry frameworks, which have traditionally been formulated for compact or bounded settings, to an important class of non-compact manifolds relevant in scattering theory and inverse problems.

Geometric Setting: Asymptotically Conic Manifolds

The authors work on smooth metrics $g$ over non-compact manifolds $M$ modeled near infinity on exact metric cones, i.e., $g \sim dr^2 + r^2 h_0(dy)$, with $(N, h_0)$ a closed Riemannian manifold. Specifically, asymptotically conic refers to manifolds for which the metric, in terms of a boundary defining function $p \sim 1/r$, has an expansion of the form $g = \frac{dp^2}{p^4} + \frac{h_p}{p^2}$ near infinity, with $h_p$ a smooth family of metrics on the boundary at infinity $\partial M = N$. This class models the geometry of non-compact perturbations of cones (including, as a special case, asymptotically Euclidean space).

The core assumption for the main results is that $(M,g)$ is non-trapping (all geodesics escape to infinity) and has no conjugate points.

X-Ray Transform and Symmetric Tensors

Following the tradition established in geometric inverse problems, the X-ray transform $I_m$ is defined on symmetric $m$-tensors by integration along complete non-trapped geodesics. The analysis includes an explicit gauge reduction to the so-called solenoidal tensors (components orthogonal to symmetrized exact cycles), as the kernel of the X-ray transform acting on $m$-tensors always contains the symmetrized differentials:

$$I_m(Dh) = 0 \text{ for } h \in C^\infty(M; S^{m-1}(T^*M)).$$

Injectivity—solenoidal injectivity—corresponds to the property that $I_m$ vanishes only on this natural kernel.

Main Results: Injectivity and Rigidity

Injectivity Theorem

The key result is Theorem 1.1, which asserts, under non-trapping and absence of conjugate points, the solenoidal injectivity of the X-ray transform on functions and symmetric tensors:

• Functions ($m=0$): If $f \in p^k C^\infty(M)$ with $I_0f=0$, then $f=0$.
• 1-forms ($m=1$): If $I_1f=0$, there exists $u \in p^{k-1}C^\infty(M)$ with $f = du$.
• General tensors: Under non-positive sectional curvature, injectivity holds for all tensor orders $m \geq 1$.

Strong numerical estimates and $L^2$-regularity bounds on the corresponding resolvent are established, providing effective control over the constructed solutions (see Lemma 3.7).

Lens Rigidity and Boundary Determination

Lens data is formalized in this non-compact setting as the pair $(S_g, L_g)$, where $S_g$ is the scattering map (input–output relation for geodesics at infinity, viewed as a symplectomorphism of $T^*\partial M$) and $L_g(\gamma)$ is a suitably renormalized geodesic length. The analysis shows that the linearized kernel of the lens rigidity map corresponds to the vanishing of the X-ray transform of certain metric derivatives, establishing a deformation rigidity statement: for negatively curved, non-trapping asymptotically conic manifolds, any smooth family of metrics with identical lens data is isometric up to a diffeomorphism fixed at infinity (Theorem 1.2).

Boundary determination results further show that, under dynamical or geometric conditions on $(\partial M, h_0)$ (e.g., ergodic geodesic flow, negative boundary curvature, or sufficiently large injectivity radius), the full boundary jet of the metric can be recovered from the scattering map $S_g$.

Analytic Methods

The paper integrates deep microlocal and geometric analytic tools:

• Normal forms and Taylor expansions for the metric near infinity allow precise control of the asymptotics and enable the jet determination arguments.
• Pestov identity for tensors on non-compact manifolds (Section 3.4.4) underpins the uniqueness proofs, supported by Carleman-type estimates and explicit control on Jacobi field growth near infinity.
• Gauge reduction gives a canonical solenoidal representative in each equivalence class. This enables the authors to exploit the integral geometry machinery in the non-compact setting.

Comparison and Contrasts

Unlike the compact boundary rigidity context, where deformation rigidity can fail for certain smooth magnetic or Finsler perturbations, the non-compact asymptotically conic setting treated here proves to be more rigid under the non-trapping/no conjugate points assumption, as there are no non-trivial (decaying) metric deformations with identical lens data.

Notably, the results are sharper than previous injectivity results on Cartan–Hadamard surfaces with decaying curvature [cf. LRS], as they hold for all functions and tensors without sign assumptions on curvature except in specific boundary cases.

Implications and Outlook

The paper’s injectivity and rigidity results have direct applications for the analysis of inverse scattering problems, where the control of wave propagation and metric recovery from scattering data fundamentally rests upon such geometric integral transforms. The explicit stability estimates, reducible to the X-ray stability theorems, are particularly relevant for quantitative inverse problems.

The geometric rigidity implications strongly constrain the possible non-trivial isospectral or isoscattering deformations within the class of asymptotically conic metrics, reinforcing the expectation that such manifolds have a unique asymptotic geometry up to isometry for fixed lens data.

Further developments could extend the methods to include metrics with mild trapping (beyond Pollicott–Ruelle resonance-free case), metrics with less regular behavior at infinity, or even vector bundle versions relevant for geometric inverse problems in gauge theory or relativity.

Conclusion

The article provides a comprehensive extension of the X-ray transform and associated rigidity theorems to asymptotically conic manifolds, demonstrating injectivity and strong deformation rigidity under natural geometric conditions. These results advance the analytic and geometric understanding of inverse problems in scattering theory and furnish new obstructions to the existence of non-isometric perturbations of the standard conic metric for large classes of non-compact manifolds (1910.09631).