2024/03/21 by Bram Mesland, Mesland, Bram, Adam Rennie +1 · 1 citation
Mathematics · Physics and Astronomy · #58B34 #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2404.07957
openalex publication_date 2024/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Using the Levi-Civita connection on the noncommutative differential one-forms of a spectral triple (\B,\H,\D), we define the full Riemann curvature tensor, the Ricci curvature tensor and scalar curvature. We give a definition of Dirac spectral triples and derive a general Weitzenbock formula for them. We apply these tools to θ-deformations of compact Riemannian manifolds. We show that the Riemann and Ricci tensors transform naturally under θ-deformation, whereas the connection Laplacian, Clifford representation of the curvature and the scalar curvature are all invariant under deformation.