Curvature and Weitzenböck formula for spectral triples

Abstract Using the Levi‐Civita connection on the noncommutative differential 1‐forms of a spectral triple , 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 ‐deformations of compact Riemannian manifolds. We show that the Riemann and Ricci tensors transform naturally under ‐deformation, whereas the connection Laplacian, Clifford representation of the curvature, and the scalar curvature are all invariant under deformation.