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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?

edited Dec 4 at 10:50

J. H. S.
2,425●12●31

asked Sep 27 at 17:09

DAVID
137●5

	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 at 17:53
	why do you say this? – DAVID Oct 8 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 at 13:25

Suhyoung Choi · Accepted Answer · 2012-11-22 11:02:20Z UTC

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.

complement of a codimension-one projective subspace

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