The first main result of this article is to prove the convergence of Lott's delocalized eta invariant holds for all invertible operators. Our second main result is to construct a pairing between delocalized cyclic cocycles of the group algebra of the fundamental group of a manifold and K-theoretic higher rho invariants of the manifold, when the fundamental group is hyperbolic. As an application, under the assumption of hyperbolicity of the fundamental group, we compute the delocalized part of the Connes-Chern character of Atiyah-Patodi-Singer type K-theoretic higher indices, for example, the higher index of a spin manifold with boundary where the boundary carries a positive scalar curvature metric. Our explicit formula for this delocalized Connes-Chern character is expressed in terms of our pairing of delocalized cyclic cocycles and higher rho invariants on the boundary of manifold. 查看全文>>