  We show that the Gaussian kernel $\exp\left\{-\lambda d_g^2(\bullet,
\bullet)\right\}$ on any non-simply-connected closed Riemannian manifold
$(\mathcal{M},g)$, where $d_g$ is the geodesic distance, is not positive
definite for any $\lambda > 0$, combining analyses in the recent preprint~[9]
by Da Costa--Mostajeran--Ortega and classical comparison theorems in Riemannian
geometry.
