2025/05/20 by Joakim Arnlind, Arnlind, Joakim, Victor Hildebrandsson +1
Computer Science · Mathematics · Physics and Astronomy · #46L87 #FOS: Mathematics #FOS: Physical sciences #Logic, programming, and type systems #Mathematical Physics (math-ph) #Mathematical and Theoretical Analysis #Noncommutative and Quantum Gravity Theories #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2505.13984
openalex publication_date 2025/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the existence of Levi-Civita connections, i.e torsion free connections compatible with a hermitian form, in the setting of derivation based noncommutative differential calculi over ∗-algebras. We prove a necessary and sufficient condition for the existence of Levi-Civita connections in terms of the image of an operator derived from the hermitian form. Moreover, we identify a necessary symmetry condition on the hermitian form that extends the classical notion of metric symmetry in Riemannian geometry. The theory is illustrated with explicit computations for free modules of rank three, including noncommutative 3-tori. We note that our approach is algebraic and does not rely on analytic tools such as C^∗-algebra norms.