  We consider the Kato and the Dynkin class and their local counterparts on a
smooth Riemannian manifold as Fr\'{e}chet spaces. Based on recent results by
Carron, Mondello and Tewodrose we show that for a Riemannian manifold $(X,g)$
of dimension $m\geq 2$ with spectral negative part $\sigma^-_g$ of the Ricci
curvature in $L^q_{\phi_g}(X,g)+L^\infty(X,g)$ for some $q>m/2$, the function
$\sigma^-_g$ is in the Kato class of $(X,g)$ if and only if $(X,g)$ satisfies a
Gaussian upper heat kernel bound for small times and is locally volume
doubling. Here $L^q_{\phi_g}(X,g)$ is the $L^q$-space which is weighted with
the inverse volume function. By establishing a localization result for the
Dynkin norm, we prove that the local Kato class and the local Dynkin class do
not depend on the chosen Riemannian metric and thus can be defined as
Fr\'{e}chet spaces on arbitrary smooth manifolds. Moreover, we prove that
smooth compactly supported functions are dense in the local Kato class and we
use this result to prove that Schr\"odinger semigroups with Kato decomposable
potentials are space-time continuous.
