[Paper Review] Lens rigidity with partial data in the presence of a magnetic field

Hanming Zhou|arXiv (Cornell University)|May 20, 2016

Numerical methods in inverse problems27 references3 citations

TL;DR

This paper establishes local lens rigidity for conformal Riemannian metrics with a magnetic field on manifolds of dimension at least three, showing that the conformal factor and magnetic field can be uniquely determined near a strictly convex boundary point from partial lens data—namely, scattering relations and travel times of unit-speed magnetic geodesics. The key result extends boundary and lens rigidity to magnetic systems under partial data and conformal invariance.

ABSTRACT

In this paper we consider the lens rigidity problem with partial data for conformal metrics in the presence of a magnetic field on a compact manifold of dimension $\geq 3$ with boundary. We show that one can uniquely determine the conformal factor and the magnetic field near a strictly convex (with respect to the magnetic geodesics) boundary point where the lens data is accessible. We also prove a boundary rigidity result with partial data assuming the lengths of magnetic geodesics joining boundary points near a strictly convex boundary point are known. The local lens rigidity result also leads to a global rigidity result under some strictly convex foliation condition. A discussion of a weaker version of the lens rigidity problem with partial data for general smooth curves is given at the end of the paper.

Motivation & Objective

• To address the lens rigidity problem for magnetic systems with partial data, particularly in the presence of a magnetic field.
• To determine whether the conformal factor and magnetic field can be uniquely recovered from incomplete boundary measurements near a strictly convex boundary point.
• To establish a boundary rigidity result under partial data assuming knowledge of magnetic geodesic lengths near a convex boundary point.
• To derive a global rigidity result under a strictly convex foliation condition.
• To explore a weaker version of the lens rigidity problem for general smooth curves using weighted X-ray transforms.

Proposed method

• Formulates the lens data as the scattering relation $L$ and travel time $ au$ for unit-speed magnetic geodesics on the unit sphere bundle over the boundary.
• Uses a magnetic flow generated by a Hamiltonian system with symplectic form $\beta = \beta_0 + \pi^*\Omega$, where $\Omega$ is a closed 2-form representing the magnetic field.
• Applies an integral identity (Equation 19) involving the logarithmic ratio of conformal factors along magnetic geodesics, derived from equality of lens data.
• Reduces the problem to a matrix-weighted X-ray transform $I_{\mathcal{W}}\varphi = 0$, where $\mathcal{W}$ is an invertible matrix field on the sphere bundle.
• Proves local invertibility of the weighted X-ray transform $I_{\mathcal{W}}$ using arguments analogous to those in geodesic flow rigidity, enabling uniqueness of the conformal factor.
• Establishes global rigidity under a strictly convex foliation condition by extending local results via propagation of singularities and microlocal analysis techniques.

Experimental results

Research questions

• RQ1 Can the conformal factor of a Riemannian metric be uniquely recovered from partial lens data in the presence of a magnetic field?
• RQ2 Under what geometric conditions does the scattering relation and travel time data uniquely determine the magnetic field and conformal factor near a boundary point?
• RQ3 How does the lens rigidity problem for magnetic systems relate to the boundary rigidity problem when only partial data is available?
• RQ4 What is the role of the weighted X-ray transform in recovering the conformal factor from lens data?
• RQ5 Can the boundary jet assumption in the rigidity result be weakened to only requiring equality on the boundary?

Key findings

• The conformal factor $c$ is uniquely determined in a neighborhood of a strictly convex boundary point if the lens data (scattering relation and travel time) agree and the boundary jets of $c$ and $\tilde{c}$ are equal.
• The magnetic field $G$ is assumed fixed in the main result, allowing recovery of the conformal factor $c$ from lens data under partial observations.
• A local lens rigidity result is established via a matrix-weighted X-ray transform $I_{\mathcal{W}}$, which is shown to be locally invertible, implying uniqueness of the conformal factor.
• A global rigidity result holds under a strictly convex foliation condition, extending the local result to the entire manifold.
• The paper provides a weaker rigidity result for general smooth curves by reducing the problem to a weighted X-ray transform, though full invertibility is not guaranteed without additional assumptions.
• The result is robust under conformal invariance and applies to manifolds of dimension $n \geq 3$ with smooth boundary.