What a terrible article. Can anyone who is not a mathematician tell me one thing they learned from this?

The naked term "manifold" in its modern usage, refers to a topological manifold, loosely a locally euclidean hausdorff topological space, which has no geometry intrinsic to it at all. The hyperbolic plane and the euclidean plane are different geometries you can put on the same topological manifold, and even does not depend on the smooth structure. In order to add a geometry to such a thing, you must actually add a geometry to it, and there are many inequivalent ways to do this systematically, none of which work for all topological manifolds.

Well as a non mathematician all I saw in your description was opaque jargon. "locally euclidean hausdorff topological space" means nothing to me. It'd be like if I asked what the Spanish word "¡hola!" meant and the answer was in evocative Spanish poetry. Extremely unlikely to be helpful to that person who doesn't know basic greetings.

This article breaks that loop and it's refreshing to see a large topic not explained as an amalgamation of arcane jargon

> Can anyone who is not a mathematician tell me one thing they learned from this?

I can share my two take-aways.

- in the geometric sense, manifolds are spaces analogous to curved 2d surfaces in 3d that extend to an arbitrary number of dimensions

- manifolds are locally Euclidean

If I were to extrapolate from the above, i'd say that:

- we can map a Euclidean space to every point on a manifold and figure out the general transformation rules that can take us from one point's Euclidean space to another point's.

- manifolds enable us to discuss curved spaces without looking at their higher-dimension parent spaces (e.g. in the case of a sphere surface we can be content with just two dimensions without working in 3d).

Naturally, I may be totally wrong about all this since I have no knowledge on the subject...

ok but she was talking about riemann