Composition of rotations and parallel transport

This note provides the details and proofs of the results announced by Levi 1993 Fields Insitute Communications vol 1 pp 133 - 8. The main result of this note is a geometrical representation of the reconstruction problem for SO(3) in terms of parallel transport. It is, of course, well known that the solution of a linear equation in cannot in general be expressed by

because the coefficient matrices and may fail to commute for . Nevertheless, when n = 3 and when is skew-symmetric, i.e. when it lies in the Lie algebra of the group of rigid rotations in , the above false formula is almost correct, as we will show here. The main result of this note is a geometrical expression for the matrix solution X(t) of matrix equations on TSO(3) of the form

where are matrices with real coefficients. The argument relies on a theorem of Poinsot together with some observations on geodesic curvatures of moving curves.