  We develop the formalism for noncommutative differential geometry and
Riemmannian geometry to take full account of the *-algebra structure on the
(possibly noncommutative) coordinate ring and the bimodule structure on the
differential forms. We show that *-compatible bimodule connections lead to
braid operators $\sigma$ in some generality (going beyond the quantum group
case) and we develop their role in the exterior algebra. We study metrics in
the form of Hermitian structures on Hilbert *-modules and metric compatibility
in both the usual and a cotorsion form. We show that the theory works well for
the quantum group $C_q[SU_2]$ with its 3D calculus, finding for each point of a
3-parameter space of covariant metrics a unique `Levi-Civita' connection
deforming the classical one and characterised by zero torsion,
metric-preservation and *-compatibility. Allowing torsion, we find a unique
connection with classical limit that is metric-preserving and *-compatible and
for which $\sigma$ obeys the braid relations. It projects to a unique
`Levi-Civita' connection on the quantum sphere. The theory also works for
finite groups and in particular for the permutation group $S_3$ where we find
somewhat similar results.
