We generalize the concept of sub-Riemannian geometry to infinite-dimensional manifolds modeled on convenient vector spaces. On a sub-Riemannian manifold $M$, the metric is defined only on a sub-bundle $\calH$ of the tangent bundle $TM$, called the horizontal distribution. Similarly to the finite-dimensional case, we are able to split possible candidates for minimizing curves into two categories: semi-rigid curves that depend only on $\calH$, and normal geodesics that depend both on $\calH$ itself and on the metric on $\calH$. In this sense, semi-rigid curves in the infinite-dimensional case generalize the notion of singular curves for finite dimensions. In particular, we study the case of regular Lie groups. As examples, we consider the group of sense-preserving diffeomorphisms $\Diff S^1$ of the unit circle and the Virasoro-Bott group with their respective horizontal distributions chosen to be the Ehresmann connections with respect to a projection to the space of normalized univalent functions. In these cases we prove controllability and find formulas for the normal geodesics with respect to the pullback of the invariant K��hlerian metric on the class of normalized univalent functions. The geodesic equations are analogues to the Camassa-Holm, Huter-Saxton, KdV, and other known non-linear PDE.

37pp