2002/11/13 by Eli Hawkins, Hawkins, Eli
Mathematics · Physics and Astronomy · #46L65 #53D17 #58B34 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Theory (hep-th) #Quantum Algebra (math.QA) #gr-qc #hep-th #math.DG #math.QA #msc:46L65 #msc:53D17 #msc:58B34
paper · pdf · doi:10.48550/arxiv.math/0211203
27 pages. Serious typo corrected from v3
arxiv created 2004/02/10 · arxiv updated 2009/11/30
Using very weak criteria for what may constitute a noncommutative geometry, I show that a pseudo-Riemannian manifold can only be smoothly deformed into noncommutative geometries if certain geometric obstructions vanish. These obstructions can be expressed as a system of partial differential equations relating the metric and the Poisson structure that describes the noncommutativity. I illustrate this by computing the obstructions for well known examples of noncommutative geometries and quantum groups. These rigid conditions may cast doubt on the idea of noncommutatively deformed space-time.