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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 $R^d$.

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    $\begingroup$ How about the Moulton plane? $\endgroup$ – Ethan MacBrough Feb 24 '17 at 17:58
  • $\begingroup$ 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? $\endgroup$ – user56834 Feb 24 '17 at 18:09
  • $\begingroup$ 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. $\endgroup$ – Ethan MacBrough Feb 24 '17 at 18:16
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A space with a Minkowski geometry is an affine space with a non euclidean geometry. In such a geometry the notion of orthogonality is defined using an ''inner product'' that is not positive defined and we have not the usual rotations but hyperbolic rotations. This is the geometry of the relativity theory.

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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).

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    $\begingroup$ 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. $\endgroup$ – user56834 Feb 24 '17 at 17:56

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