  We find a unique torsion free Riemannian spin connection for the natural
Killing metric on the quantum group $C_q[SL_2]$, using a recent frame bundle
formulation. We find that its covariant Ricci curvature is essentially
proportional to the metric (i.e. an Einstein space). We compute the Dirac
operator and find for $q$ an odd $r$'th root of unity that its eigenvalues are
given by $q$-integers $[m]_q$ for $m=0,1,...,r-1$ offset by the constant
background curvature. We fully solve the Dirac equation for $r=3$.
