Noncommutative Gravity

Various approaches by the author and collaborators to define gravitational fluctuations associated with a noncommutative space are reviewed. Geometry of a noncommutative space is defined by the data (A, H,D) where A is a noncommutative involutive algebra, H is a separable Hilbert space and D a self-adjoint operator on H referred to as Dirac operator [1]. Geometry on Riemannian manifolds could be recovered by specializing to the data A = C (M) , H = L(S), D = γ ( ∂μ + 1 4 ω ab μ γab ) , where ω ab μ is the spin-connection on a manifold M. To deserve the name geometry the operator D should satisfy certain conditions [2]. Presented at TH-2002, Paris, France, July 2002.