Paolo Aschieri

We consider the problem of Levi-Civita connections of arbitrary metrics on noncommutative spaces. 

After reviewing the triangular quantum group and the associated quantum (homogeneous) algebras cases we study the case of arbitrary Hopf algebras. 

In the context of Woronowicz bicovariant differential calculi we show how connections on one forms or vector fields extend to the braided symmetric tensor product of forms or vector fields. 

This allows to define a metric compatibility condition between an arbitrary connection (not necessarily a bimodule connection) and an arbitrary braided symmetric metric. 

The metric compatibility condition and the torsion free condition are solved for metrics conformal to central and equivariant metrics provided the braiding given by the differential calculus is diagonalizable (and an associated map invertible). 

Thus, existence and uniqueness of the Levi-Civita connection on a quantum group for this class of metrics that are neither central nor equivariant is proven. 

This includes the SL_q(2) example.