- A manifold can be thought of as a set of coordinate systems such that any point in one coordinate system can be made the origin of coordinates in other coordinate systems of the set. The application of a manifold is that if one observer sets up coordinates to study space or space time at one origin, while another does the same thing somewhere else, and then the structure of the manifold describes the relationships between points in one set of coordinates and those of the other. Manifolds inherit many of the local properties of Euclidean space. But if parallel lines are drawn in one coordinate system, and extended across others, they do not necessarily remain parallel.
I moved this paragraph here because while I think our manifold article needs a first paragraph explaining the use and ideas behind manifolds for non-specialists, the above paragraph doesn't do the job:
- a manifold is not a set of coordinate systems: a manifold is some "curved" space or surface where coordinate systems can be introduced locally. The space or surface is the central object; the coordinate systems are just crutches.
- space time is conceptualized as a manifold, and that definitely should be stated.
- the parallel line statement has to do with Riemannian manifolds; you can't talk about parallel lines in manifolds unless you have a concept of distance and angle.
Perhaps someone could add a definition for paracompact? I assume I'm not the only one who's clueless in this regard... --Belltower