Tomographic Fourier extension identities for submanifolds of $${\mathbb {R}}^n$$

We establish identities for the composition \(T_{k,n}(|\widehat{gd\sigma }|^2)\), where \(g\mapsto \widehat{gd\sigma }\) is the Fourier extension operator associated with a general smooth k-dimensional submanifold of \({\mathbb {R}}^n\), and \(T_{k,n}\) is the k-plane transform. Several connections to problems in Fourier restriction theory are presented.