Hawkins, Eli The structure of noncommutative deformations. (English) Zbl 1130.53062 J. Differ. Geom. 77, No. 3, 385-424 (2007). The author continues his studies on geometrical structures associated with noncommutative deformations of Riemannian manifolds, originated in an earlier paper [Commun. Math. Phys. 246, No. 2, 211–235 (2004; Zbl 1055.58001)]. He proves that one obstruction for such a deformation is the vanishing of a certain rank 5 tensor (called metacurvature) constructed out of a flat contravariant connection associated with the induced Poisson bracket on the Grassmann algebra of differential forms. Together with the obstructions considered in the previous paper this leads to a structure consisting of a Riemannian tensor and a Poisson tensor satisfying certain compatibility conditions. It is proved that, in the case of a compact Riemannian manifold, the Poisson tensor can be constructed locally from commuting Killing vectors and a general construction of geometric deformations for any compatible Poisson structure is sketched. Finally, a few examples of compatible Poisson structures on Riemannian manifolds are given. Reviewer: Janusz Grabowski (Warszawa) Cited in 3 ReviewsCited in 15 Documents MSC: 53D55 Deformation quantization, star products 81R60 Noncommutative geometry in quantum theory 58B34 Noncommutative geometry (à la Connes) 53D17 Poisson manifolds; Poisson groupoids and algebroids Keywords:Riemannian metric; Poisson tensor; connection; divergence; spectral triples; associative algebra deformation Citations:Zbl 1055.58001 × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid