Existence and uniqueness of the Levi-Civita connection on noncommutative differential forms
Bram Mesland, Adam Rennie
TL;DR
This work advances noncommutative Riemannian geometry by establishing a complete criterion for the existence of Hermitian torsion-free connections on the bimodule of differential one-forms Ω^1_d(B) within a broad first-order calculus framework. It translates the problem into a two-projection problem on a higher tensor power and provides explicit constructions, notably for θ-deformations of compact Riemannian manifolds, yielding a unique Hermitian torsion-free bimodule connection and thereby a noncommutative Levi-Civita connection. The results recover the classical Levi-Civita connection on manifolds and extend the theory to isospectral deformations, braided two-tensor structures, and nonunital/indefinite settings, often with explicit connection forms. The paper thus unifies Connes-inspired differential calculus, Hilbert C*-module techniques, and braiding concepts to produce concrete, computable Levi-Civita-type objects in noncommutative geometry with potential broad applicability to quantum manifolds and deformations.
Abstract
We combine Hilbert module and algebraic techniques to give necessary and sufficient conditions for the existence of an Hermitian torsion-free connection on the bimodule of differential one-forms of a first order differential calculus. In the presence of the extra structure of a bimodule connection, we give sufficient conditions for uniqueness. We prove that any $θ$-deformation of a compact Riemannian manifold admits a unique Hermitian torsion-free bimodule connection and provide an explicit construction of it. Specialising to classical Riemannian manifolds yields a novel construction of the Levi-Civita connection on the cotangent bundle.