NON-COMMUTATIVE SURFACES FORMALISM AND APPLICATION TO 2D GRAVITY

Abstract

In this project, the main goal is to be familiarized with the formalism of non-commutative geometry. This work develops non-commutative deformations of Riemannian geometry in the light of Whitney theorem. Two steps have been covered toward this goal: 1.Introducing the Moyal algebra A, which is a non-commutative deformation of the algebra of smooth functions on a region of R 2 ,and 2.Development of the non-commutative Riemannian geometry for two dimensional surfaces embedded in three dimensional space. Application is made for spherical surfaces to obtain the elements of the 2D non-commutative gravity, i.e. metric, left connection, Riemannian tensor, Ricci tensor and scalar curvature. Key words : Non-commutative surfaces; Connection; Riemannian and Ricci tensors.