Comment by yatopifo

> 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...