The main result of this note is a new formula which establishes the connection between two the concept of parallel transport on the sphere on the one hand and the composition of orthogonal matrices on the other.
Viewed alternatively, this simple-looking formula establishes the equivalence of two problems: (i) recovering a curve on the unit sphere from its geodesic curvature (given as a function of the arclength) and (ii) finding the composition of a family of non-communting matrices, or more precisely, solving the linear matrix equation X'=A(t) X where X and A are 3-by-3 matrices, A(t) is skew-symmetric and X is in SO(3).