  We generalize the geodesic rule to the case of formation of higher
codimensional global defects. Relying on energetic arguments, we argue that,
for such defects, the geometric structures of interest are the totally geodesic
submanifolds. On the other hand, stochastic arguments lead to a diffusion
equation approach, from which the geodesic rule is deduced. It turns out that
the most appropriate geometric structure that one should consider is the convex
hull of the values of the order parameter on the causal volumes whose collision
gives rise to the defect. We explain why these two approaches lead to similar
results when calculating the density of global defects by using a theorem of
Cheeger and Gromoll. We present a computation of the probability of formation
of strings/vortices in the case of a system, such as nematic liquid crystals,
whose vacuum is $\mathbb{R}P^2$.
