ar X iv : 0 81 0 . 23 57 v 1 [ m at h - ph ] 1 4 O ct 2 00 8 PROJECTIVE MODULE DESCRIPTION OF EMBEDDED NONCOMMUTATIVE SPACES

Noncommutative differential geometry over the Moyal algebra is developed following an algebraic approach. It is then applied to investigate embedded noncommutative spaces. We explicitly construct the projective modules corresponding to the tangent bundles of the noncommutative spaces, and recover from this algebraic formulation the metric, Levi-Civita connection and related curvature introduced in earlier work. Transformation rules of connections and curvatures under general coordinate changes are given explicitly. A bar involution on the Moyal algebra is discovered, and its consequences on the noncommutative differential geometry are described.