Sharp Hardy and Rellich type inequalities on Cartan--Hadamard manifolds and their improvements

In this paper, we prove several new Hardy type inequalities (such as the weighted Hardy inequality, weighted Rellich inequality, critical Hardy inequality and critical Rellich inequality) for radial derivations (i.e., the derivation along the geodesic curve) on Cartan--Hadamard manifolds. By Gauss lemma, our new Hardy inequality are stronger than the classical one. We also established the improvements of these inequalities in terms of sectional curvature of underlying manifolds which illustrate the effect of curvature to these inequalities. Furthermore, we obtain some improvements of Hardy and Rellich inequality in hyperbolic space $\mathbb H^n$. Especially, we show that our new Rellich inequality is indeed stronger the classical one in hyperbolic space $\mathbb H^n$.