Riemannian Geometry of Noncommutative Surfaces

Abstract

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.

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

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Cited by in corpus (20)

• Emergent Geometry and Gravity from Matrix Models: an Introduction
• Corrections to Schwarzschild Solution in Noncommutative Gauge Theory of Gravity
• Noncommutative Geometries and Gravity
• Geodesic equation in non-commutative gauge theory of gravity
• Noncommutative gauge theory using covariant star product defined between Lie-valued differential forms
• Gauging the twisted Poincare symmetry as noncommutative theory of gravitation
• Exact solutions of noncommutative vacuum Einstein field equations and plane-fronted gravitational waves
• Quantum deformations of Schwarzschild and Schwarzschild-de Sitter spacetimes
• On the Newtonian limit of emergent NC gravity and long-distance corrections
• Breakdown of Lorentz Invariance for Spin-1/2 Particle Motion in Curved Space-Time with Applications to Muon Decay
• Analogy between the Schwarzschild solution in a noncommutative gauge theory and the Reissner-Nordström metric
• Gravitational collapse of spherically symmetric stars in noncommutative general relativity
• Higher derivative Chern-Simons extension in the noncommutative QED${}_3$
• Signature Change by GUP
• Curvature and geometric modules of noncommutative spheres and tori
• Projective module description of embedded noncommutative spaces
• A noncommutative geometric approach to the quantum structure of spacetime
• Holographic Entanglement Entropy for noncommutative Anti-de Sitter space
• Noncommutative corrections to the minimal surface areas of the pure AdS spacetime and Schwarzschild-AdS black hole
• On 2d gravity and parallel transport field equation on quantum sphere