Riemannian geometry of noncommutative surfaces

Journal Article · Tue Jul 15 04:00:00 EDT 2008 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.2953461· OSTI ID:21100341

Chaichian, M; Tureanu, A ^[1]; Zhang, R B ^[2]; Xiao, Zhang ^[3]

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1. Department of Physical Sciences, University of Helsinki and Helsinki Institute of Physics, P.O. Box 64, 00014 Helsinki (Finland)
2. School of Mathematics and Statistics, University of Sydney, Sydney, New South Wales 2006 (Australia)
3. Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190 (China)

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.

• OSTI ID: 21100341
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 7 Vol. 49; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
COMMUTATION RELATIONS
COORDINATES
GENERAL RELATIVITY THEORY
GEOMETRY
QUANTUM GRAVITY
SPHERES
SURFACES
TRANSFORMATIONS