  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.
