When I say "SR" in this post, I mean the set of classical and quantum theories of particles and fields in Minkowski spacetime. I'm trying to come up with a list of topics in SR that can be dealt with in a better way when we have defined Minkowski spacetime as a manifold instead of as a vector space (or an affine space). Maybe there aren't that many? The ones I can think of right away are Born rigidity (I've seen a definition that uses Lie derivatives, and I haven't seen one that would work with a spacetime that doesn't have a manifold structure) A classification of types of fields we might be interested in. (Sections of various vector bundles over Minkowski spacetime). The solid rotating disc, if we need to analyze it much more deeply than anyone wants to (except one person I know ) Definitions of measurable quantities that we'd prefer to be explicitly coordinate independent (i.e. proper time). A coordinate independent definition of geodesics and inertial motion. That's pretty much it. Maybe the Lagrangian/Hamiltonian stuff is more natural in this context too? Let me know if you can think of other stuff that you think is easier to explain or can be treated in a more general or more elegant way in the "manifold version" of the theory than in the "vector space version". Also let me know if you think the stuff I've mentioned can be handled just as well in a vector space.