Star product geometries

We consider noncommutative geometries obtained from a triangular Drinfeld twist. This allows us to construct and study a wide class of noncommutative manifolds and their deformed Lie algebras of infinitesimal diffeomorphisms. Principles of symmetry can be implemented in this way. Two examples are considered: (a) the general covariance in noncommutative spacetime, which leads to a noncommutative theory of gravity, and b) symplectomorphims of the algebra of observables associated with noncommutative configuration spaces, which leads to a geometric formulation of quantization on noncommutative spacetime (i.e., we establish a noncommutative correspondence principle from ⋆-Poisson brackets to ⋆-commutators).