Curvature and Weitzenbock formula for spectral triples

Abstract

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 $\theta$-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.