Ignoring the boundary term in the action is appropriate only for manifolds that are both compact and without boundary, as stated in the discussion. The application of Stokes' theorem indicates that the integral vanishes if the manifold has no boundary, but it must also be compact for the theorem to hold. In general relativity, most useful spacetimes are non-compact and without boundary, such as Minkowski spacetime, which requires the inclusion of the Gibbons-Hawking boundary term to derive the Einstein field equations correctly. The conversation also highlights that the choice of manifold type affects the resulting equations and solutions, emphasizing the importance of understanding the topology involved. Ultimately, the discussion underscores the need to carefully consider boundary conditions in the variational principle applied to the action in general relativity.