Endow the real projective plane with the distance defined by $d(L,L')$ := "the angle between the lines $L$ and $L'$ ".


It is the case that every isometry from $RP^2$ onto $RP^2$ is induced by an element of the orthogonal group $O(3)$.


I want to use this fact in a research paper. But it turns out that all my considerations are so elementary that, apart from this fact, this paper is readable by a 3rd year student at the University. I would like to have a reference for this result that would be readable by such a student (without resorting to Riemannian structures).


But I have no reference at all for this result actually, so a reference with more fancy technology would already be valuable.


Thank you very much.



Elementary reference for the isometry group of $\mathbb{RP}^2$
Robert Bryant's post provides a neat proof: it is a good reference for the result I am interested in. I found another (more conventional: published book vs MO answer) elementary reference for it: 9.7.1 and 19.1.2.2 in Berger's *Geometry*. This proof is less trickily efficient than that of Robert Bryant, but it is also interesting (geometric insight). I have decided to cite both.


