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

Abstract

Given a tame differential calculus over a noncommutative algebra $\mathcal{A}$ and an $\mathcal{A}$-bilinear pseudo-Riemannian metric $g_0,$ consider the conformal deformation $g=k.g_0,$ $k$ being an invertible element of $\mathcal{A}.$We prove that there exists a unique connection $\nabla$ on the bimodule of one-forms of the differential calculus which is torsionless and compatible with $g.$ We derive a concrete formula connecting $\nabla$ and the Levi-Civita connection for the pseudo-Riemannian metric $g_0.$ As an application, we compute the Ricci and scalar curvature 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.

Most of the results were included in arxiv 1606.08142 which has now been split into two parts. New results include a short proof for existence of Levi-Civita connections on conformally deformed metrics and an equivalent criterion of metric-compatibility of a connection on tame differential calculi