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" the complement of a codimension-one projective subspace of RP3(real projective space) is identifiable in a geodesic-structure manner with an affine 3-space so that the group of projective transformations acting on it is identical with the group of affine transformations of the affine 3-space.we call this set an ' affine patch ' .conversely , a natural completion of an affine 3-space is identified with RP3 in a geodesic preserving manner ".

what does 'geodesic preserving manner' mean? I know it means that we can embed affine 3-space in RP3 by a map which preserves geodesics , but how does this map do this?

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    Do you actually want to know what the map is? I am not sure what you mean by "how does this map to this". Can you give the source from which you quoted the description?2012-10-10
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    let me ask another question?2012-10-11
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    how can we embed an affine space in a vector space?2012-10-11
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    is this related to your question above at all? $\mathbb{R}P^3$ is not a vector space. The point is that this embedding is not fully structure preserving: it is not preserving the affine structure of $\mathbb{E}^3$. I'm pretty sure this map is just an embedding in the sense of differential topology. It has the additional property that geodesics are preserved, but nothing is said about affine, or Riemann structure in addition.2012-10-12
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    Seems david has asked [this](http://mathoverflow.net/questions/108266) [question](http://mathoverflow.net/questions/108789) before. The source appears to be [this paper](http://www.mathnet.or.kr/mathnet/kms_tex/980027.pdf) by Suhyoung Choi.2012-10-12

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