  We study the differential and Riemannian geometry of algebras $A$ endowed
with an action of a triangular Hopf algebra $H$ and noncomutativity compatible
with the associated braiding. The modules of one forms and of braided
derivations are modules in a compact closed category of $H$-modules
$A$-bimodules, whose internal morphisms correspond to tensor fields. Different
approaches to curvature and torsion are proven to be equivalent by extending
the Cartan calculus to left (right) $A$-module connections. The Cartan
structure equations and the Bianchi identities are derived. Existence and
uniqueness of the Levi-Civita connection for arbitrary braided symmetric
pseudo-Riemannian metrics is proven.
