Curvature and Weitzenböck formula for spectral triples

Using the Levi-Civita connection on the noncommutative differential 1-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 Weitzenböck formula for them. We apply these tools to $\theta$-deformations of compact Riemannian manifolds. We show that the Riemann and Ricci tensors transform naturally under $\theta$-deformation, whereas the connection Laplacian, Clifford representation of the curvature, and the scalar curvature are all invariant under deformation.