Levi-Civita connections for conformally deformed metrics on tame differential calculi

Given a tame differential calculus over noncommutative algebra [Formula: see text] and an text]-bilinear metric consider the conformal deformation being invertible element of We prove that there exists unique connection on bimodule one-forms which is torsionless compatible with derive concrete formula connecting Levi-Civita for As application, we compute Ricci scalar curvatures general perturbation canonical text]-torus as well natural quantum Heisenberg manifold. For latter, curvature turns out to be negative constant.