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Let $A$ be a real affine space with $\text{dim}(A)>1$ and $f:A\rightarrow A$ be an arbitrary map such that: for each affine line $L\subset A$, there exists an affine line $N\subset A$ such that $f(L)\subseteq N$; and for each set $L\subseteq A$ which is not a subset of an affine line, there is no affine line $N\subset A$ such that $f(L)\subseteq N$. In other words, let $f$ preserve colinearity and non-colinearity. Does this imply that $f$ is an affine transform?

I am aware that this is clearly not true in case $n=1$, but for other cases I couldn't prove it nor find a counter example. Does anyone have any helping idea?

cnikbesku
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  • Are you familiar with the theorem that this is true for bijective maps (sometimes called the "fundamental theorem of affine geometry")? – Eric Wofsey Nov 13 '23 at 15:22
  • @EricWofsey I am not, but thank you! I can look it up now. EDIT: I actually seemingly can't find anything similar. Can you please add some helping detail on how to look it up? – cnikbesku Nov 13 '23 at 22:30
  • See for instance https://math.stackexchange.com/questions/2035287/is-a-function-which-preserves-zero-and-affine-lines-necessarily-linear. Unfortunately there is not any obvious way to adjust the proof without the bijection hypothesis. In particular, the proof makes heavy use of the fact that $f$ will preserve parallel lines, which follows from surjectivity but does not follow from just your assumptions in any way that I can see. – Eric Wofsey Nov 13 '23 at 22:41
  • @EricWofsey Thank you for providing me with that! I believe that theorem 1.8 in this article: https://arxiv.org/pdf/2212.02429.pdf answers my question (I found it on the page you mentioned). I am still unsure about what exactly "maps lines onto lines" is supposed to mean. – cnikbesku Nov 13 '23 at 23:00
  • The author in that paper on the page 6 at the beginning of step 2 skips the part of proving that such function preserves parallelism. This is exactly the part you said is cruical so I am getting a reason to doubt the proof. – cnikbesku Nov 13 '23 at 23:16
  • I would assume "maps lines onto lines" means the image of any line is a line. In particular this includes a surjectivity assertion (the restriction to a line is surjective onto the image line), which is enough to get that parallel lines are preserved. – Eric Wofsey Nov 13 '23 at 23:29
  • @EricWofsey Alright, now it makes a lot of sense. I guess this finalizes my research on this, thank you for your assistance. – cnikbesku Nov 14 '23 at 18:00

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