X-ray transform on Damek-Ricci spaces. (English) Zbl 1254.44002
Summary: Damek-Ricci spaces, also called harmonic \(NA\) groups, make up a large class of harmonic Riemannian manifolds including all hyperbolic spaces. We prove here an inversion formula and a support theorem for the X-ray transform, i.e. integration along geodesics, on those spaces.
Using suitably chosen totally geodesic submanifolds we reduce the problems to similar questions on low-dimensional hyperbolic spaces.
Using suitably chosen totally geodesic submanifolds we reduce the problems to similar questions on low-dimensional hyperbolic spaces.
MSC:
44A12 | Radon transform |
53C22 | Geodesics in global differential geometry |
53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
22E30 | Analysis on real and complex Lie groups |