  We provide sharp forms of $k$-plane transform inequalities on the
$d$-dimensional sphere $\mathbb{S}^d$ and the $d$-dimensional hyperbolic space
$\mathbb{H}^d$. In particular, we prove that extremizers do not exist for
$\mathbb{H}^d$. This work is a natural extension of some results for the
$k$-plane transform on $\mathbb{R}^d$.
