Levi-Civita connections for a class of spectral triples

We give a new definition of Levi-Civita connection for a noncommutative pseudo-Riemannian metric on a noncommutative manifold given by a spectral triple. We prove the existence-uniqueness result for a class of modules of one-forms over a large class of noncommutative manifolds, including the matrix geometry of the fuzzy 3-sphere, the quantum Heisenberg manifolds and Connes-Landi deformations of spectral triples on the Connes-Dubois- Violette-Rieffel deformation of a compact manifold equipped with a free toral action. It is interesting to note that in the example of the quantum Heisenberg manifold, the definition of metric compatibility given in Frohlich et al. (Commun Math Phys 203:119-184, 1999) failed to ensure the existence of a unique Levi-Civita connection. In the case of the matrix geometry, the Levi-Civita connection that we get coincides with the unique real torsion-less unitary connection obtained by Frohlich et al. (1999).