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