  We review the circumstances under which test particles can be localized
around a spacetime section \Sigma_0 smoothly contained within a codimension-1
embedding space M. If such a confinement is possible, \Sigma_0 is said to be
totally geodesic. Using three different methods, we derive a stability
condition for trapped test particles in terms of intrinsic geometrical
quantities on \Sigma_0 and M; namely, confined paths are stable against
perturbations if the gravitational stress-energy density on M is larger than
that on \Sigma_0, as measured by an observed travelling along the unperturbed
trajectory. We confirm our general result explicitly in two different cases:
the warped-product metric ansatz for (n+1)-dimensional Einstein spaces, and a
known solution of the 5-dimensional vacuum field equation embedding certain
4-dimensional cosmologies. We conclude by defining a confinement energy
condition that can be used to classify geometries incorporating totally
geodesic submanifolds, such as those found in thick braneworld and other
5-dimensional scenarios.
