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

TL;DR

This paper establishes the existence and explicit form of Levi-Civita connections for conformally deformed metrics on tame noncommutative differential calculi, and computes curvature invariants for specific noncommutative spaces.

Contribution

It provides a unique torsionless, metric-compatible connection formula for conformally deformed metrics in noncommutative geometry, extending classical Levi-Civita theory.

Findings

Derived explicit connection formula for conformally deformed metrics.

Computed Ricci and scalar curvature for noncommutative 2-torus.

Found scalar curvature to be a negative constant on the quantum Heisenberg manifold.

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.