  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.
