Curvature of differentiable Hilbert modules and Kasparov modules

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Advances in Mathematics


In this paper we introduce the curvature of densely defined universal connections on Hilbert C⁎-modules relative to a spectral triple (or unbounded Kasparov module), obtaining a well-defined curvature operator. Fixing the spectral triple, we find that modulo junk forms, the curvature only depends on the represented form of the universal connection. We refine our definition of curvature to factorizations of unbounded Kasparov modules. Our definition recovers all the curvature data of a Riemannian submersion of compact manifolds, viewed as a KK-factorization.

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