  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\"ahlerian
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.
