Existence and uniqueness of the Levi-Civita connection on noncommutative differential forms
- Resource Type
- Working Paper
- Authors
- Mesland, Bram; Rennie, Adam
- Source
- Subject
- Mathematics - Quantum Algebra
Mathematical Physics
Mathematics - Differential Geometry
Mathematics - Operator Algebras
58B34
- Language
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 $\theta$-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.
Comment: 53 pages. Typos corrected and references updated. New proof to Lemma 6.3