Riemannian geometry of noncommutative surfaces

A Riemannian geometry of noncommutative n-dimensional surfaces is developed as a first step toward 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 analog of the first and second Bianchi identities. As examples, noncommutative analogs 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 to other treatments