Sharp Poincar\'e-Hardy and Poincar\'e-Rellich inequalities on the hyperbolic space

We study Hardy-type inequalities associated to the quadratic form of the shifted Laplacian $-\Delta_{\mathbb H^N}-(N-1)^2/4$ on the hyperbolic space ${\mathbb H}^N$, $(N-1)^2/4$ being, as it is well-known, the bottom of the $L^2$-spectrum of $-\Delta_{\mathbb H^N}$. We find the optimal constant in the resulting Poincar\'e-Hardy inequality, which includes a further remainder term which makes it sharp also locally. A related inequality under suitable curvature assumption on more general manifolds is also shown. Similarly, we prove Rellich-type inequalities associated with the shifted Laplacian, in which at least one of the constant involved is again sharp.