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We prove \emph{optimal} improvements of the Hardy inequality on thehyperbolic space. Here, optimal means that the resulting operator is criticalin the sense of [J.Funct.Anal. 266 (2014), pp. 4422-89], namely the associatedinequality cannot be further improved. Such inequalities arise from moregeneral, \emph{optimal} ones valid for the operator $ P_{\lambda}:=-\Delta_{\mathbb{H}^N} - \lambda$ where $0 \leq \lambda \leq\lambda_{1}(\mathbb{H}^N)$ and $\lambda_{1}(\mathbb{H}^N)$ is the bottom of the$L^2$ spectrum of $-\Delta_{\mathbb{H}^N} $, a problem that had been studied in[J.Funct.Anal. 272 (2017), pp. 1661-1703 ] only for the operator$P_{\lambda_{1}(\mathbb{H}^N)}$. A different, critical and new inequality on$\mathbb{H}^N$, locally of Hardy type, is also shown. Such results have in factgreater generality since there are shown on general Cartan-Hadamard manifoldsunder curvature assumptions, possibly depending on the point.Existence/nonexistence of extremals for the related Hardy-Poincar\'einequalities are also proved using concentration-compactness technique and aLiouville comparison theorem. As applications of our inequalities we obtain animproved Rellich inequality and we derive a quantitative version ofHeisenberg-Pauli-Weyl uncertainty principle for the operator $P_\lambda.$

An optimal improvement for the Hardy inequality on the hyperbolic space and related manifolds