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

Article Type

Research Article

Publication Title

International Journal of Mathematics


Given a tame differential calculus over a noncommutative algebra and an-bilinear metric g0, consider the conformal deformation g = k.g0, k being an invertible element of . We prove that there exists a unique connection on the bimodule of one-forms of the differential calculus which is torsionless and compatible with g. We derive a concrete formula connecting and the Levi-Civita connection for the metric g0. As an application, we compute the Ricci and scalar curvatures for a general conformal perturbation of the canonical metric on the noncommutative 2-Torus as well as for a natural metric on the quantum Heisenberg manifold. For the latter, the scalar curvature turns out to be a negative constant.



Publication Date



Open Access, Green

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