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Let $f:\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ ($n \geq 2$) be a diffeomorphism which preserves lines (i.e. the image of any line is a line). Prove that $f$ is an affine map.

The idea is to prove that the Jacobian matrix is constant and I can finish from there, but I do not see how to use the condition that $f$ sends lines to lines. Thanks!

jjagmath
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Steppewolf
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    The Jacobian matrix is the matrix of partial derivatives. Try to relate partial derivatives to the inclination of the lines $f(L_i)$, where $L_i$ is the line corresponding to the $x_i$-axis. This should suffice to show that the Jacobian is constant. – deabo Sep 16 '22 at 01:52
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    You can write $i$-th partial derivative at $x$ as $\frac d{dt} \left.f(x+te_i)\right|_{t=0}$ and all points $x+te_i$ are on the same line. – Ennar Sep 16 '22 at 02:06
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    I think that the question is rather subtle, since the local version of this claim does not hold. (If you have a diffeomorphism between open subsets of $\mathbb R^n$ that maps lines to lines, then it is the restriction of a projective transformation and not necessarily of an affine transformation.) So a purely local proof cannot work. As far as I know the simplest way to prove this is to first prove the local theorem for open subsets of $\mathbb RP^n$ (with projective transformations) and then show that a projective transformation of $\mathbb RP^n$ which maps $\mathbb R^n$ to itself is affine. – Andreas Cap Sep 18 '22 at 07:50

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