Riemannian Geometry of Noncommutative Surfaces

Details Riemannian Geometry of Noncommutative Surfaces Riemannian Geometry of Noncommutative Surfaces

2006-12-13

arxiv.org/abs/hep-th/0612128v2

J.Math.Phys.49:073511,2008

Physics

High Energy Physics

High Energy Physics - Theory

28 pages, some clarifications, examples and references added, version to appear in J. Math. Phys

Scientific paper

10.1063/1.2953461

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.

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