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

We show that the Gaussian kernel $\exp \{-\lambda d_g^2(\bullet ,\bullet )\}$ 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.