  We consider the group of sense-preserving diffeomorphisms $\Diff S^1$ of the
unit circle and its central extension, the Virasoro-Bott group, with their
respective horizontal distributions chosen to be Ehresmann connections with
respect to a projection to the smooth universal Teichm\"uller space and the
universal Teichm\"uller curve associated to the space of normalized univalent
functions. We find formulas for the normal geodesics with respect to the
pullback of the invariant K\"ahlerian metrics, namely, the Velling-Kirillov
metric on the class of normalized univalent functions and the Weil-Petersson
metric on the universal Teichm\"uller space. The geodesic equations are
sub-Riemannian analogues of the Euler-Arnold equation and lead to the CLM, KdV,
and other known non-linear PDE.
