  A (positive) locally convex curve in the 2-sphere is a curve with positive
geodesic curvature (i.e., which always turns left). In the 3-sphere, it is a
curve with positive torsion. In this work we discussed the topology of spaces
of such curves with prescribed initial and final jets. The case of the 2-sphere
is understood (Saldanha-2013); the case of n=3 is not yet thoroughly clarified.
In order to obtain partial information about the homotopy type of such spaces
in the case n=3, we represented a positive locally convex curve as a pair of
curves on the 2-sphere with some restrictions.
