  A Riemannian geometry of noncommutative n-dimensional surfaces is developed
as a first step towards the construction of a consistent noncommutative
gravitational theory. Historically, as well, Riemannian geometry was recognized
to be the underlying structure of Einstein's theory of general relativity and
led to further developments of the latter. The notions of metric and
connections on such noncommutative surfaces are introduced and it is shown that
the connections are metric-compatible, giving rise to the corresponding Riemann
curvature. The latter also satisfies the noncommutative analogue of the first
and second Bianchi identities. As examples, noncommutative analogues of the
sphere, torus and hyperboloid are studied in detail. The problem of covariance
under appropriately defined general coordinate transformations is also
discussed and commented on as compared with other treatments.
