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

Sub-Riemannian structures corresponding to K\"ahlerian metrics on the universal Teichmueller space and curve