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The complement of a codimension-one projective subspace of $\mathbb{R}\mathbf{P}^{3}$ is identifiable in a geodesic structure preserving 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.

In the above sentence what does the "geodesic structure preserving manner" part mean?

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The intersection of each geodesic in $\mathbb{R}P^3$ with the complement of the subspace is either empty or is identified with a line in the affine 3-space. –  Lee Mosher Sep 27 '12 at 17:53
    
why do you say this? –  DAVID Oct 8 '12 at 6:55
    
DAVID -- "in a geodesic preserving manner" means that one can identify the complement of a hyperplane with an affine space so that each geodesic in the projective space, which is a line, goes to a geodesic in the affine space, also a line. I.e., the metrics are different (one comlpete, one not) but their geodesics coincide. –  algori Nov 22 '12 at 13:25

1 Answer 1

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Hi. One can read this in M. Berger's Geometry I, Chapter 5, from Springer Verlag. This is standard classical work. An affine space can be completed to a projective space by adding a subspace of codimension-one and a real projective space becomes an affine space by removing a subspace of codimension-one.

Also, please see my new MSJ book No. 27. One can find it in my homepage and http://mathsoc.jp/publication/memoir/memoirs-e.html. It has some explanations.

Anyway you can send me questions also.

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in your article " geometric structures on low-dimensional manifolds " section - 1. theorem - 3 , I really don't understand it. why we need to show that around the edges the identifications give us trivial holonomy elements? and why you said " at each sphere the triangulations are from points of RP2 " ? and then you concluded that " at each sphere and the vertex on it , the holonomy around the vertex is the identity map " ?? –  DAVID Dec 3 '12 at 7:03

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