  We present a generic functional inequality on Riemannian manifolds, both in
additive and multiplicative forms, that produces well known and genuinely new
Hardy-type inequalities. For the additive version, we introduce Riccati pairs
that extend Bessel pairs developed by Ghoussoub and Moradifam (Proc. Natl.
Acad. Sci. USA, 2008 & Math.A nn., 2011). This concept enables us to give very
short/elegant proofs of a number of celebrated functional inequalities on
Riemannian manifolds with sectional curvature bounded from above by simply
solving a Riccati-type ODE. Among others, we provide alternative proofs for
Caccioppoli inequalities, Hardy-type inequalities and their improvements,
spectral gap estimates, interpolation inequalities, and
Ghoussoub-Moradifam-type weighted inequalities. Concerning the multiplicative
form, we prove sharp uncertainty principles on Cartan-Hadamard manifolds, i.e.,
Heisenberg-Pauli-Weyl uncertainty principles, Hydrogen uncertainty principles
and Caffarelli-Kohn-Nirenberg inequalities. Some sharpness and rigidity
phenomena are also discussed.
