×

The structure of noncommutative deformations. (English) Zbl 1130.53062

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.

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

Citations:

Zbl 1055.58001