# Example of an affine space that is not euclidean

Wikipedia says that Euclidean space is affine (obvious), but that not all affine spaces are Euclidean.

I understand that Euclidean space has extra structure defined on it, namely metrics of distance and angles.

However, are there actual examples of affine spaces that are not Euclidean?

ps. I am not talking about the vector space $R^d$, but about the manifold $E^d$, which is a metric affine space, but not a vector space, though it can be charted by the cartesian coordinates in $R^d$.

• How about the Moulton plane? – Ethan MacBrough Feb 24 '17 at 17:58
• Indeed. Wow, ok that is very weird. I'm having a hard time seeing why this is affine, but I'll look at it some more. Is there a more intuitive example? – user56834 Feb 24 '17 at 18:09
• It's probably possible to construct a "more intuitive" example, but this was the simplest example for which I could find a Wiki page with an image on it. – Ethan MacBrough Feb 24 '17 at 18:16

An euclidean space is an affine space which contain $0$.
So for instance the line of equation $y=1$ in the plane in an affine space, but it is not an euclidean space ($0$ does not belong to this line).
• I don't think this is correct. You are implicitly saying that $E^d =R^d$, but $R^d$ in that sense is technically only the cartesian coordinate system of the manifold $E^d$. $R^d$ is a vector space, but $E^d$ is not. Maybe you think this is nitpicking (or that I am wrong), but it is essential to my question, because otherwise indeed I would have thought of that example. – user56834 Feb 24 '17 at 17:56