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

03/12/2023

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by   Siran Li, et al.

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We show that the Gaussian kernel exp{-λ d_g^2(∙, ∙)} on any non-simply-connected closed Riemannian manifold (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.