Higher-Rank Radon Transforms on Constant Curvature Spaces

Abstract

We study higher-rank Radon transforms of the form f (τ ) → τ ⊂ζ f (τ ), where τ is a j-dimensional totally geodesic submanifold in the n-dimensional real constant curvature space and ζ is a similar submanifold of dimension k > j. The corresponding dual transforms are also considered. The transforms are explored the Euclidean case (affine Grassmannian bundles), the elliptic case (compact Grassmannians), and the hyperbolic case (the hyperboloid model, the Beltrami-Klein model, and the projective model). The main objectives are sharp conditions for the existence and injectivity of the Radon transforms in Lebesgue spaces, transition from one model to another, support theorems, and inversion formulas. Conjectures and open problems are discussed.