Curvature and Weitzenböck formula for spectral triples

Bram Mesland and Adam Rennie

Mathematische Nachrichten, 2024, vol. 297, issue 12, 4582-4604

Abstract: Using the Levi‐Civita connection on the noncommutative differential 1‐forms of a spectral triple (B,H,D)$(\mathcal {B},\mathcal {H},\mathcal {D})$, 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 Weitzenböck formula for them. We apply these tools to θ$\theta$‐deformations of compact Riemannian manifolds. We show that the Riemann and Ricci tensors transform naturally under θ$\theta$‐deformation, whereas the connection Laplacian, Clifford representation of the curvature, and the scalar curvature are all invariant under deformation.

Date: 2024
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https://doi.org/10.1002/mana.202400158

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