  We show how to define Riemannian metrics and connections on a noncommutative
torus in such a way that an analogue of Levi-Civita's theorem on the existence
and uniqueness of a Riemannian connection holds. The major novelty is that we
need to use two different notions of noncommutative vector field. Levi-Civita's
theorem makes it possible to define Riemannian curvature using the usual
formulas.
