Sharp uncertainty principles on Riemannian manifolds: the influence of curvature

We present a rigidity scenario for complete Riemannian manifolds supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in $\mathbb R^n$ (shortly, sharp HPW principle). Our results deeply depend on the curvature of the Riemannian manifold which can be roughly formulated as follows: (a) When $(M,g)$ has non-positive sectional curvature, the sharp HPW principle holds on $(M,g)$. However, positive extremals exist in the sharp HPW principle if and only if $(M,g)$ is isometric to $\mathbb R^n$, $n={\rm dim}(M)$. (b) When $(M,g)$ has non-negative Ricci curvature, the sharp HPW principle holds on $(M,g)$ if and only if $(M,g)$ is isometric to $\mathbb R^n$. Since the sharp HPW principle and the Hardy-Poincar\'e inequality are endpoints of the Caffarelli-Kohn-Nirenberg interpolation inequality, we establish further quantitative results for the latter inequalities in terms of the curvature on Cartan-Hadamard manifolds.