  The free complex sphere $S^{N-1}_{\mathbb C,+}$ is the noncommutative
manifold defined by the equations $\sum_ix_ix_i^*=\sum_ix_i^*x_i=1$. Certain
submanifolds $X\subset S^{N-1}_{\mathbb C,+}$, related to the quantum groups,
are known to have Riemannian features, including an integration functional. We
review here the known facts on the subject.
