T. Kato was a pioneer of the spectral theory of Schrödinger operators with singular potentials in Euclidean spaces. His great achievements rely on several celebrated tools and a lot of work has been done to transplant them to Riemannian manifolds. In this talk we shall focus on the so called “Lp Positivity Preservation” and discuss on how it is (un)related to the geometry of the underlying space. The talk is based on a joint work with Daniele Valtorta and Giona Veronelli.