∗∗-compatible connections in noncommutative Riemannian geometry

We develop the formalism for noncommutative differential geometry and Riemmannian geometry to take full account of the ∗∗-algebra structure on the (possibly noncommutative) coordinate ring and the bimodule structure on the differential forms. We show that ∗∗-compatible bimodule connections lead to braid operators σσ in some generality (going beyond the quantum group case) and we develop their role in the exterior algebra. We study metrics in the form of Hermitian structures on Hilbert ∗∗-modules and metric compatibility in both the usual form and a cotorsion form. We show that the theory works well for the quantum group Cq[SU2]Cq[SU2] with its three-dimensional calculus, finding for each point of a three-parameter space of covariant metrics a unique ‘Levi-Civita’ connection deforming the classical one and characterised by zero torsion, metric preservation and ∗∗-compatibility. Allowing torsion, we find a unique connection with a classical limit that is metric preserving and ∗∗-compatible and for which σσ obeys the braid relations. It projects to a unique ‘Levi-Civita’ connection on the quantum sphere. The theory also works for finite groups, and in particular for the permutation group S3S3, where we find somewhat similar results.