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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 positivedefinite for any $\lambda > 0$, combining analyses in the recent preprint~[9]by Da Costa--Mostajeran--Ortega and classical comparison theorems in Riemanniangeometry.

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