A smooth curve in an n-dimensional Riemannian manifold M is called free of order k if its first k covariant derivatives are linearly independent at each point. Free curves of order smaller than n were studied by Smale, Feldman and Gromov, who showed that they satisfy the h-principle. In particular, when k < n, the space of all based free curves of order k is homotopy equivalent to the loop space on the bundle of k-dimensional frames in the tangent bundle of M.
When k = n, the h-principle fails to be true already for the simplest examples. I will discuss these examples and show that a somewhat weaker statement holds: for any n > 2 the loop space on the bundle of n-frames in TM is the group completion of the monoid of free curves of order n in M.