  We consider a length functional for $C^1$ curves of fixed degree in graded
manifolds equipped with a Riemannian metric. The first variation of this length
functional can be computed only if the curve can be deformed in a suitable
sense, and this condition is expressed via a differential equation along the
curve. In the classical differential geometry setting, the analogous condition
was considered by Bryant and Hsu in [Invent. Math., 114(2):435-461, 1993, J.
Differential Geom., 36(3):551-589, 1992], who proved that it is equivalent to
the surjectivity of a holonomy map. The purpose of this paper is to extend this
deformation theory to curves of fixed degree providing several examples and
applications. In particular, we give a useful sufficient condition to guarantee
the possibility of deforming a curve.
