Gaussian kernels on nonsimply connected closed Riemannian manifolds are never positive definite

Abstract

AbstractWe show that the Gaussian kernel on any nonsimply connected closed Riemannian manifold , where is the geodesic distance, is not positive definite for any , combining analyses in the recent preprint  by Da Costa–Mostajeran–Ortega and classical comparison theorems in Riemannian geometry.