Gaussian kernels on non-simply-connected closed Riemannian manifolds are never positive definite

We show that the Gaussian kernel $\exp\left{-\lambda dg^2(\bullet, \bullet)\right}$ on any non-simply-connected closed Riemannian manifold $(\mathcal{M},g)$, where $dg$ 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.