  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.
