CURVATURE FOR DIFFERENTIABLE HILBERT MODULES

In this talk, we introduce the curvature of densely defined universal connections on Hilbert C*-modules, relative to a spectral triple, leading to a well defined curvature operator. Algebraically, this curvature can be interpreted as the defect of the unbounded Kasparov product to commute with the operation of taking squares. The definition recovers the represented curvature of finitely generated projective modules as well as all the curvature data of a Riemannian submersion of compact manifolds, viewed as a KK-factorization.