Steven Flynn

Tensor Tomography on groups of Heisenberg-type

Groups of Heisenberg type (H-type) are a large family of Lie groups diffeomorphic to Euclidean space, whose harmonic analysis is essentially governed by the Schrodinger representation. They carry a natural left-invariant metric and horizontal distribution, giving them the structure of a torus bundle with a connection. H-type groups also furnish examples of non-isometric isospectral Riemannian manifolds, providing counterexamples to the nonlinear inverse problem "can one hear the shape of a drum."

We study the following linear inverse problem: when can one determine a function or symmetric tensor field on an H-type group from its integrals over horizontal (sub-Riemannian) geodesics? The operator which assigns to a function its integrals over such geodesics is called the X-ray transform. We generalize previous results for the X-ray transform on the Heisenberg group, to this large family of groups. This is part of a program to study horizontal tensor tomography on Carnot groups (and their nilmanifolds), the local models for sub-Riemannian manifolds.