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.

翻译：本文结合Da Costa、Mostajeran与Ortega近期预印本[9]中的分析以及黎曼几何中的经典比较定理，证明在任意非单连通闭黎曼流形$(\mathcal{M},g)$上，高斯核$\exp\left\{-\lambda d_g^2(\bullet, \bullet)\right\}$（其中$d_g$为测地距离）对于任意$\lambda > 0$均非正定。