Definition talk:Riemannian Manifold

$\R^n$ is the image of coordinate functions which for each point on the manifold produce $n$ numbers known as the coordinates (c.f. Definition:Chart)

What do you mean? I still cannot understand the meaning of "on $\R^n$". It looks simply wrong. I guess it means it is a $\R$-manifold and not a $\C$-manifold, or something similar. --Usagiop (talk) 00:23, 10 November 2022 (UTC)

Probably we mean the same thing. $\R^n$ should mean real locally Euclidean space.--Julius (talk) 08:06, 10 November 2022 (UTC)

While a manifold is defined as a "topological space", I wonder whether that is as helpful as emphasising that it is in fact a vector space, and that its elements are vectors. That seems to have been taken for granted. Perhaps it is the locally euclidean space which should be identified as being a vector space. Can someone clarify this? --prime mover (talk) 07:13, 9 August 2023 (UTC)