# complement of a codimension-one projective subspace

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