### Abstract

We present a theoretical study of the director fields in toroidal geometries. We find a spontaneous chiral symmetry breaking: despite the achiral nature of nematics these toroids show a handedness if the toroid is thick enough, or alternatively, if the ratio $\left(K_2 - K_{24}\right)/K_3$ is small enough. The chiral-achiral transition falls in the universality class of order-disorder transition of the mean-field Ising model. The critical exponent relating the degree of twist scales with $\left(K_2 - K_{24}\right)/K_3$ and the toroidal aspect ratio is $1/2$. Remarkably, an external field does not break the chiral symmetry explicitely, but shifts the transtion. In the case of toroidal cholesterics, we do find explicit chiral symmetry breaking. The critical exponant relating the degree of twist to the helicity is $1/3$. Finally, combining these results with recent experimental investigations of toroidal droplets of 5CB yields an estimate of the saddle-splay constant, $K_{24}\approx K_2$.