  We make some observations about Rosenberg's Levi-Civita connections on
noncommutative tori, noting the non-uniqueness of torsion-free
metric-compatible connections without prescribed connection operator for the
inner *-derivations, the nontrivial curvature form of the inner *-derivations,
and the validity of the Gauss-Bonnet theorem for two classes of non-conformal
deformations of the flat metric on the noncommutative two-tori, including the
case of non-commuting scalings along the principal directions of a two-torus.
