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

2021/01/18 by Jyotishman Bhowmick, Bhowmick, Jyotishman, Debashish Goswami +3
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2101.07221

openalex publication_date 2021/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a tame differential calculus over a noncommutative algebra\n\A and an \A-bilinear pseudo-Riemannian metric g₀,\nconsider the conformal deformation g = k. g₀, k being an invertible\nelement of \A.We prove that there exists a unique connection\n\∇ on the bimodule of one-forms of the differential calculus which is\ntorsionless and compatible with g. We derive a concrete formula connecting\n\∇ and the Levi-Civita connection for the pseudo-Riemannian metric g₀.\nAs an application, we compute the Ricci and scalar curvature for a general\nconformal perturbation of the canonical metric on the noncommutative 2-torus\nas well as for a natural metric on the quantum Heisenberg manifold. For the\nlatter, the scalar curvature turns out to be a negative constant.\n

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