Noncommutative differential geometry and braided commutative Riemannian geometry

In his celebrated article from 1989 Woronowicz introduced a covariant differential calculus on Hopf algebras, generalizing the Cartan calculus on Lie groups. The aim of this talk is to recall his construction and to explain how other notions of differential geometry, mainly connections, curvature and torsion, generalize to this setting. In the second half of the presentation we specialize to noncommutative spaces with triangular Hopf algebra symmetry and provide existence and uniqueness of a Levi-Civita connection for any non-degenerate metric. The latter observation is due to the speaker. As our main example we mention Drinfel'd twist deformation quantization of Riemannian geometry on Poisson manifolds.