  We prove multipolar Hardy inequalities on complete Riemannian manifolds,
providing various curved counterparts of some Euclidean multipolar inequalities
due to Cazacu and Zuazua [Improved multipolar Hardy inequalities, 2013]. We
notice that our inequalities deeply depend on the curvature, providing
(quantitative) information about the deflection from the flat case. By using
these inequalities together with variational methods and group-theoretical
arguments, we also establish non-existence, existence and multiplicity results
for certain Schr\"odinger-type problems involving the Laplace-Beltrami operator
and bipolar potentials on Cartan-Hadamard manifolds and on the open upper
hemisphere, respectively.
