  We prove second and fourth order improved Poincar\'e type inequalities on the
hyperbolic space involving Hardy-type remainder terms. Since theirs l.h.s. only
involve the radial part of the gradient or of the laplacian, they can be seen
as stronger versions of the classical Poincar\'e inequality. We show that such
inequalities hold true on model manifolds as well, under suitable curvature
assumptions and sharpness of some constants is also discussed.

Functional Analysis;Analysis of PDEs