  We consider curves $\gamma : [0, 1]\to\mathbb{R}^3$ endowed with an adapted
orthonormal frame $r : [0, 1]\to SO(3)$. We are interested in the cases where
the frame is constrained, in the sense that one of its `curvatures' (i.e.,
off-diagonal elements of $r'r^T$) is prescribed. One example is the Fr\'enet
frame. In order to deform such constrained framed curves without spoiling the
constraint, we proceed in two stages. First we deform the frame $r$ in a way
that is naturally compatible with the differential constraint, by interpreting
it in terms of parallel transport on the sphere. The deformation of the base
curve $\gamma$ is achieved in a second step, by means of a suitable
reparametrization of the frame. We illustrate this deformation procedure by
providing some applications.
