CYCLIC ROTATIONS, CONTRACTABILITY AND GAUSS-BONNET

Let a rigid object or frame of reference have identical initial and fi nal orientations but be rotated in any way in between, with an angular velocity omega(t). Any unit vector u(t) carried with the frame passes through a cycle of directions enclosing a solid angle Omega. The full relation between these three quantities is shown to be 2 pi n = Omega + integral omega . u dt, mod4 pi, where the rum number n is zero if t he sequence of orientations of the frame is contractible and unity if it is non-contractible. The main derivation uses the Calugareanu relat ion, Lk = Wr + T omega, between linking number, writhe, and twist of a ribbon loop. An outline alternative derivation uses the Berry phase o f a quantum spin 1/2. Finally the result is applied to the standard pa rallel transport holonomy expressed in the Gauss-Bonnet theorem: it is refined to be correct mod4 pi rather than merely mod 2 pi.