Radon transform and curvature

We interpret the setting for a Radon transform as a submanifold of the space of generalized functions, and compute its extrinsic curvature: it is the Hessian composed with the Radon transform. 1. The general setting. Let M and Σ be smooth finite dimensional manifolds. Let m = dim(M). A linear mapping R : C∞ c (M)→ C∞(Σ) is called a (generalized) Radon transform if it is given in the following way: To each point y ∈ Σ there corresponds a submanifold Σy of M and a density μy on Σy, and the operator R is given by R(f)(y) := ∫