  By introducing the concept of \emph{Kato control pairs} for a given
Riemannian minimal heat kernel, we prove that on every Riemannian manifold
$(M,g)$ the Kato class $\mathcal{K}(M,g)$ has a subspace of the form
$\mathsf{L}^q(M,d\varrho)$, where $\varrho$ has a continuous density with
respect to the volume measure $\mu_g$ (where $q$ depends on $\dim(M)$). Using a
local parabolic $\mathsf{L}^1$-mean value inequality, we prove the existence of
such densities for every Riemannian manifold, which in particular implies
$\mathsf{L}^q_{loc}(M)\subset\mathcal{K}_{loc}(M,g)$. Based on previously
established results, the latter local fact can be applied to the question of
essential self-adjointness of Schr\"odinger operators with singular magnetic
and electric potentials. Finally, we also provide a Kato criterion in terms of
minimal Riemannian submersions.
