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I want to find a differentiable $n$-dimensional compact manifold $M$ which can be endowed with an affine structure but cannot be endowed with a euclidean structure.

An affine (resp. euclidean) structure is a geometric structure with $X=\Bbb R^n$ and $G$ is the group of affine (resp. euclidean) transformations of $\Bbb R^n$.

I would like to find such a manifold for every possible dimension $n\geq 1$. I know that in dimension $1$ and $2$, such a manifold doesn't exist, since the only affine manifolds in that case are the circle, the torus and the Klein bottle and all of them are euclidean manifolds.

In dimension $3$, I think that $S^1\times S^2$ is an example. It is an affine manifold since it is diffeomorphic to the quotient $$\Bbb R^3-0/x\sim 2x.$$ However I don't really know how to prove that $S^1\times S^2$ has no euclidean structure (maybe we can use some theorem of Thurston about geometries of $3$-manifolds but it seems to be a "big tool")

In dimension $n\geq 4$, maybe $S^1\times S^{n-1}\simeq \Bbb R^n-0/x\sim 2x$ could be an example, but again I don't know how to prove that this manifold doesn't admit a euclidean structure.

Is there an elementary proof that these compact affine manifolds don't have euclidean structures? Are there some better examples?

Thanks in advance.