On two notions of torsion and metric compatibility of connections in noncommutative geometry

We compare the notions of metric compatibility and torsion of a connection in the frameworks of Beggs–Majid and Mesland–Rennie. It follows that for * -preserving connections, compatibility with a real metric in the sense of Beggs–Majid corresponds to Hermitian connections in the sense of Mesland–Rennie. If the calculus is quasi-tame, the torsion zero conditions are equivalent. A combination of these results proves the existence and uniqueness of Levi-Civita connections in the sense of Mesland–Rennie for unitary cocycle deformations of a large class of Riemannian manifolds, as well as the Heckenberger–Kolb calculi on all quantized irreducible flag manifolds.