Some Hardy and Rellich type inequalities for affine connections

In this article we study various forms of the Hardy inequality for affine connections on a complete noncompact Riemannian manifold, including the two-weight Hardy inequality, the improved Hardy inequality, the Rellich inequality, the Hardy-Poincaré inequality and the Heisenberg-Pauli-Weyl inequality. Our results improve and include many previously known results as special cases.