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I have a three dimensional manifold where negatives are identified, so $x = -x$, but $x$ does not equal $cx$ unless $x$ is $1$ or $-1$. Does anyone know what this manifold is? Other than the identification of the negatives, it should locally act like $\mathbb{R}^{3}$.

Edit: the manifold is $\mathbb{R}^{3}$, but with an isometry between $x = [x1, x2, x3]$ and $x = [-x1, -x2, -x3$].

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Your descriptio of the manifold is so incomplete that it is impossible to tell. –  Mariano Suárez-Alvarez Feb 15 '13 at 22:18
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If you take the $3$-sphere $S^3\subset \mathbb R^4$ and identify antipodal points, you obtain $\mathbb P\mathbb R^3$, three-dimensional projective space.

If you take all of $\mathbb R^3$ and identify each point $x$ with its negative $-x$, then watch out what happens at the origin ...

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If I ignore everything going to hell at the origin, can I compute distances just by taking D(x,y) = min(d(x,y), d(x,-y)), where d is euclidean? –  user62484 Feb 15 '13 at 22:27
    
@user62484 Yes, this is the quotient metric; the group is small enough that long chains are not required. –  user53153 Feb 15 '13 at 22:35
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